i^ 


. -/^f 


GIFT  OF 
Prof.    C.    A.    Kofold 


IDABOLL'S 

SCHOOLMASTER'S  ASSISTANT 

IMPROVED.AND  ENLARGED, 

BKING  \  VLAr:^ 

PRACTICx\L  SYSTEM 

OF 

A  R  I  T  M  BI  E  T  I  C," 

ADAPTED  TO  THE  UJNJTED  STATES. 

BY  NATHAN  DAEOLL. 


T%7T//  TU^  ADDITION  OP  ^ 

O         -  -      J       J      3         •  "     "^     " 

,        ,     ^  'J  •     J     o     «.  .-       -^    •  0 

PRACTICAL-  A€eOUNT ANT ; 

FARMERS'  AND  .MECHANICS' 

BEST  METHOD  OF  BOOK-KEEPING; 

FOR  THE  EASY  JNSTKUCTiON  OF  YOUTH 

DESIGNED 

AS  A  COMPANION'  TO  DABOLL  3  AaiTHxMETIC 
BY  SAMUEiT  GREEN 

STEREOTYPE    PRINT, 


NEIV'LONDC/N  I 
i'RlNTED  A^^D  PUBLISHED  BY  SAMUEL  GREEN, 

rroprUiorof  the  Copij-RigJif. 


BE  IT  KEMEMBERED,  That  on  th«  eleYEntb  c^  a?  1^^ 
ry,  in  the  forty-ninth  year  of  the  ind^pendeace  of  thft  Uni4 
flISttes  of  America,  Samuel  Greek-,  of  said  Distxict,  b&th  d^osi* 
ted  in  t^U  ofQce,  tfee  title  of  a  book,  the  rig-ht  where- 
of he  cla^im^  as  pfpftVi^t^ro  \^  the  v/ords  following, 
to  wit :  "  jBiibalrs.SchoijlmdStejr'??  Assistant ;  improv- 
ed and  entarged.  Being  a  plain  practical  System  of 
Arithmetic.  ad*iRl«4't'0  tne  Uni^t^d  States.  By  Na» 
thai?  BafeoU.— Whh,thc  addition  of  the:  Practical  Ac- 
or  yafi.n4r^'*aOijf  Mechanics'  bes*  metljod  of  Book-keep- 
ing ;  for  the  easy  instruction  of  youth.  Desired  as  a  Compajo* 
ton  t^Daboji's  Arithmetic.  By  Samuel  Green.  Stereotype  Pnnt." 
In  conformity  to  the  act  of  Congress  of  the  United  States,  enti- 
tled, '*  An  act  for  the  encouragement  of  learning,  by  securing  th» 
copies  of  maps,  charts,  and  books  to  the  authors  and  propnctora 
ef  them,  during  the  time'^  th  reia  mentioned. 

CHARLES  A.  INGERSOLL, 

Clerk  of  the  District  of  Connccti^ui, 
A  trttc  QOpy  o    record.     Examined  and  sealed  by  me. 
CHARLES  A.  INGERSOLL, 

Clerk  of  the  Disti  id  of  GonnecUe^g, 


countant 


^  / 


^ 


«ReOMMENT)ATI05;S, 


-  -^f*-- 


YALE-COIXEGE,   NOV,   27,    1799. 

I  HAVE  read  Daroll's  Schoolmaster's  Assistant. 
The  arrangement  of  tlie  different  branches  of  Arithmetic 
i^  judicious  and  perspicuous.  The  author  has  well  ex- 
plained Decimal  Arithmetic,  and  has  applied  it  in  a  plain 
and  elegant  manner  in  the  solution  of  various  questions, 
and  especially  to  those  relative  to  the  Federal  Computa- 
tion of  monej.  I  think  it  will  be  a  very  useful  book  to 
Schoolmasters  and  their  pupils. 

JOSIAH  MEIGS,  Professor  of 
Mathematics  and  JVdtural  Philosophy^ 
[Now  Surveyor  General  of  the  United  States.] 

X  HAVE  given  some  attention  to  the  work  above  men- 
tioned, and  concur  with  Mr.  Professor  Meigs  in  his  opin? 
ion  of  its  merit.  NOAH  WEBSTER. 

Ifew-Haven^  December  12, 1799. 


RHODE-ISLAND    COLLEGE,    NOV.    30,    1799. 

1  HAVE  run  through  Mr.  Daboll's  Schoolmaster's 
Assistant,  and  have  formed  of  it  a  very  favourable  opin- 
ion. According  to  its  original  design,  I  think  it  well 
^*  calculated  to  iurnish  Schools  in  general  with  a  method- 
ical, easy  and  comprehensive  System  of  Practical  AriUi- 
mfitic.'*  I  therefore  hope  it  may  find  a  generous  patron- 
^^y  and  have  an  extensive  spread  - 

ASA  MESSER,  Professor  of  the 
Le(irned  Lan^ua^es^  and  Teacher  of  Mathematics 
'NRw  President  of  that  Institution.'] 


^ 

::0,  1802.         I 

■SSTSTANT,       I 


-2-    MAJ.  -^SSTSTANT, 

in  teac:  ^meac,  :uid  think  it  the  best 

calciilat  ff  any  v/Jiich  lias  falleu  vrithiu 

my  obstru-k:;  JOHN  AiUMS,  Rector  of 

Fltiinfield  Jlcad&my 
rNo\v  Frmcipal  of  Piiillips  A.«  :\  '■  udovet-j  Mass."^ 


231LLERICA    ACADE^iY.  ■  '  :.C.    10,    ISOT. 

Haying  examined  Mr.  Da  Aritli- 

metic,  i  am  pleased  with  the  juG;^:neMt  ;!i;5playe(l  in  his 
method  J  and  the  perspicuitj  of  his  explanations,  and 
thinking^  it  as  easy  txvA  comprehensive  a  sjsteiu  as  anjr 
with  which  I  am  acquainted,  can  cheertiilij  recommend 
it  to  the  natronage  of  Instiuclors. 

SAMUEL  WHITING, 
Taacher  cf  Mathematics, 


FR0M    MR.    K7-X\-r.r;Y,    TF,AniJER  OF    MATHEMATICS. 

1   BEC  ■  :     _  .-,.  ^3   ScHooi.MAs- 

TE.E.'::  AssisTAN'?  on  examining  it 

attentively,  gave  ... .  >.ce  to  any  otlier 

system  extant,  a i  -/{A  ed  i  t  UiV^i e  pupi ! s 

under  my  chari^e  ^  «*.>.  .'....  ...,^^  ;L...ie  havAiusW  it  exclu- 
sively in  elementary  tuition,  to  tliegrcti'^  advantage  and 
improvement  ol'l' '"  ^^* '^ -nt,  as  well  as  the  ease  and  as- 
sistance of  the  I  I  also  deem  ^t  equally  well 
calculated  for  the  ^  . -d  of  individu:)' ^  in  private  in- 
struction ;  and  think  it  my  duty  to  give  tlie  labour  and 
ingenuity  of  the  a.H lu  r  the  tribute  of  my  hearty  approval 
and  recommendation. 

ROGF!{  KENNEDY 


-i.  lU'j  design  of  tLIs  work  is  to  fun 
tho  United  States  with  a  methodical  and  comprehensive 
system  of  Practical  Jlrithmetic^  in  '•*  hich  I  have  endea- 
voure<l,  through  the  whole,  to  have  the  rules  as  concise 
and  fiimiliar,  as  the  nature  of  the  subject  will  permit. 

During  the  long  period  which  I  have  devoted  to  the 
instruction  of  youth  in  Aritlimetic,  I  have  made  use  ot 
various  systems  which  have  just  claims  to  scientific  mer- 
it ;  but  the  authors  appear  to  have  been  deficient  in  an 
important  point — the  practical  teacher's  experience. — 
The  J  have  been  too  sparing  of  examples,  especially  in 
the  first  rudiments;  in  consequence  of  v/hich,  the  young 
pupil  is  hurried  through  tiie  ground  rules  too  fast  for  his 
capacity.  This  objection  1  have  endeavoured  t»  obviate 
in  the  following  ti*eatise. 

In  teaching  the  first  rules,  I  have  found  it  best  to  en- 
courage the  attention  of  scholars  by  a  variety  of  easy  and 
familiar  questions,  which  might  serve  to  strengthen  their 
minds  as  their  studies  grow  more  arduous. 

The  rules  are  arranged  in  such  order  as  to  introduce 
the  most  simple  and  necessary  narts,  previous  to  thos* 
which  are  more  abstruse  and  diiiicult. 

To  enter  into  a  detail  of  the  whole  work  would  be  te- 
dious ;  I  shall  therefore  notice  only  a  few  particulars,  and 
refer  the  reader  to  the  contents. 

Although  the  Federal  Coin  is  purely  decimal,  it  is  8% 
nearly  allied  to  whole  numbers,  and  so  absolutely  neces- 
sary to  be  understood  by  every  one,  that  I  have  intro- 
duced it  immediately  after  addition  ot  whole  numbers, 
and  also  shown  how  to  find  the  value  of  goods  therein, 
iimnedfetely  after  simple  multiplication  5  which  may  be 
of  gi-eat  adyanta<je  to  many,  who  perhaps  will  not  iiavc 
an  opportunity  of  learning  fractions. 

In  the  arrangement  of  fractions,  i  have  taken  an  entire 
new  method,  the  advantages  and  fi^cility  of  which  will 
iufficiently  apologiaie  far  it$  not  F  einfc  nw^rclfi^  to  other 


sYr,if r.is.  As tleciiaial  fractions raay  be leavneu i^.vacli easier 
tiian  vulgar,  and  are  more  simpiej  useful,  anil  neces- 
sary, and  soonest  v^anted  in  more  useful  brandies  of  ■ 
Arithmetic,  thej  ought  to  bfe  learned  iirst,  and  Vulgar 
Fractlon^j  omicted,  until  further  prop;ress  in  iha  science 
shall  make  tliem  necessary.  It  may  be  well  to  obtain  a 
general  idea  of  them,  and  to  attend  to  two  or  three  easy 
prol)lems  therein  :  after  which,  t!ie  scholar  may  'ieaia 
decimals,  which  will  be  necessary  in  the  reduction  of  cur- 
rencies, computing  interest  rnd r«iany  other  biancliCs. 

Besides,  to  obtain  a  thorouj^h  knowledge  of  Vulgar 
Fractions,  is  generally  a  task  too  hard  for  young  scholars 
who  have  made  no  further  progress  in  Arithmelic  tlusu 
Reduction,  and  often  discourages  them. 

I  have  therefore  placed  a  few  problems  in  Fractiom*, 
according  to  tlic  method  above  hinted  ;  and  after  going 
t?irought1ie  principal  mercantile  rules,  have  treated  upon 
Vulgar  Fractions  at  large,  the  scholar  being  now  capable 
of  going  through  them  with  advantage  and  ease. 

Li  vSimple  Interest,  in  Federal  Money,  I  have  given 
several  new  and  concise  rules  ;  some  of  which  are  par- 
ticularly designed  for  the  use  of  the  compting-house. 

The  •Appendix  contains  a  variety  of  rules  for  casting 
Interest,  Rebate,  &c.  together  with  a  nuriiber  of  the  most 
easy  and  useful  problems,  for  measuring  superficies  and 
solids,  examples  of  forms  commnnly  used  in  transacting 
business,  useful  tables,  &c.  which  are  designed  as  aids  iJi 
the  common  business  of  life.  ^ 

Perfect  accuracy,  in  a  work  of  this  nature,  can  hardly 
be  expected  ;  errors  of  the  press,  or  perhaps  of  tlie  au- 
thor, may  have  escaped  correction.  If  any  such  arc  point 
ed  out,  it  will  be  conxsislered  as  a  mark  of  frieiK.; ship  and 
favor,  by 

The  public^ s  most  hnmhle 

and  obedient  Servant^ 

NATHAN  DAIUJLL. 


TABLE  Oy  CONTE>i;TS, 

Addition,  simple 

^ of  Federal  Money    . 

. Compound 

Alligation 

Annuities  or  Pensions,  at  Compound  Interest 

Arithmetical  progression 

Barter         ...-.♦ 

Brokerage       .         .  .      .        .► 

Characters,  Explanation  of 

Commission 

Conjoined  Proportion 

Coins  of  the  United  States,  Weights  of 

Division  of  "Whole  Nunibei's 

Contections  in 

Compound 

Discount 

Duodecimals       •        .        .        . 
Equation  of  Payments 
Evolution,  or  Extraction  of  Roots 
Exchange         .         •         .         • 
Federal  Money 


Fellowship 


Subtraction  o! 


Compound 


Fractions,  Vulgar  and  Decimal 
Insurance  .... 
fnterest.  Simple 

—  by  Decimals 

Compound 

bv  Decimals 


Inverse  Proportion 
Involution 
Loss  and  Gain 
Multiplication, 


Simple    . 

Application  and  use 
Supplimentto 
Conipouiiii 


Kurneratior 

Practice 

Position 

Vf^vmutatlon  of  Quantities 


Qf 


17 


57 

135 

£28 

1S8 

179 

151 

21 

27 

144 

146 

74,  155 

126 

129 

169 

154 

177 

167 

178 

HO 

23 

S3 

SS 

51 

15 

109 

200 

20 


Cy.l  TABLE   OF   CONTENTS, 

Questf .« 's  lor  exercise         .        .        ,        *        .  209 

Ileduttion       ...*....  65 

— —  of  Currencies,  do.  of  Coin  .  89,  95 

Rule  of  Tliree  Direct,  do.  Inverse           .            100,  108 

Double 148 

Rules,  for  reducing  thedificrent  currencies  of  the 
several  United  States,  also  Canada  and  No- 
va-Scotia, eacli  to  the  par  of  all  others         96,  97 

• — — Application  of  the  preceding      ...  98 

Short  Practical,  for  calculating  Interest  126 

for  casting  Interest  at  6  per  cent.      .         .  215 

•- ^-  for  finding  the  contents  oj  Superfices  &  Solids  220 

— — -to  reduce  the  currencies  of  the  different 

States,  to  Federal  Money          .         .         .  218 
Rebate,  A  sliort  method  of  fm^ding  tlie,  of  any  giv- 
en sum  for  months  and  days         .        .  217 
Subtraction,  Simifle              .         T        -         .         •  25 

, —  Compound           ....  45 

Table,  Numeration  and  Pence     ....  9 

•™ —  Addition,  Subtraction,  and  Multiplication  10 

of  Weight  and  Measure             .         .         •  11 

•— —  oF  I'ime  and  Motion     ....  15 

« -  showing  i\\Q^  number  of  days  from  any  day 

of  one  month,  to  the  same'day  in  any  other 

monlii       . 172 

Fhowing  the  amount  uf  iLor  1  dolhir,  at  5  & 

G  per  cent  Compound  Interest, for  20  years  232 
*-™-  showing  theamount  of  1Z.  annuity,  foroome 
for  51  jeiirs  or  under,  at  5  and  6  per  cent. 

Compound  Interest           .        .         .    '     •  233 
-— -  shov.-nig  the  present  worth  of  IJ.  annuity,  for 

SI  yrs.  at  5  &i  6  per  c.  Compound  Interest  ib. 
.«-^ — ^  of  cents,  ansv/ering  to  the  currencies  of  the 


United  States,  with  Sterling,  he.      .         .  256 
-— - —  showing  the  value  of  Federal  Money  in 

other  currencies        .         .         .         .  *      .  237 

Tai-e  and  Trett 1 14 

Useful  Fonns  in  transacting  business           .        .  238 
Weiglits  of  several  pieces  of  English,  Portuguese, 

h  French,  gokl  coins,  in  dollars,  cts.  &  mtlhs  2S4 

—  --of  English ^^  Portu«vuese  gold,      do.      do.  2So 

of  French  twA  Spamsh  iJ!;old,         city,      d^  ib. 


jdhu^  CMoA^llu. 


X>,lIjp^L'b 


SCHOOLMASTER'S  ASSISTANT 


M>4^^« 


ARITHMETICAL  TABLES. 


lVumreai07i  TdbU 

Pence  Tfl&Ze. 

flO 

£?. 

5. 

d. 

d. 

s. 

«} 

■§ 

20 

Is  1 

8 

12  U  1 

1 

C3 

tft 

SO 

2 

C 

24 

2 

, 

3 
O 

•§ 

40 

3 

4 

S6 

3 

1 

£ 

^ 

g 

50 

4 

2 

48 

4 

«ib4 

13J 

Cm 

g 

60 

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0 

60 

5 

o 

CO 

£ 

§ 

52 

o 
en 

§ 

o 

1 

O 
Pi 

o 

3 

"5 
1 

1 

43 

QO 

H 

TO 
80 
90 

too 

110 

5 
6 
7 
8 
9 

10 
8 
6 
4 
2 

72 

84 

96 

108 

120 

6 

»r 

8 

9 

10 

i9 

8 

/ 

6 

5 

4 

3 

2 

1 

120 

10 

0 

1S2 

U 

9 

8 

7 

6 

5 

4 

S 

2 

9 

B 
9 

7 
8 
9 

6 
7 
8 
9 

5 

6 

7 
8 

4 
5 
6 

7 

3 
4 
5 
6 

•ti 

niGti 

e 

9 

8 

^ 

4  faj-tlungs  1 

pctujV/ 

f/ 

9 

8 

V2  pence., 

1    i 

jhillingj 

^ 

9 

m  bit 

iliiii. 

^,1 

nouiui,/. 

/. 

APl^UTXW/'Ai?li   SUS'I'RACTION  TABLE. 


T\\:  dU:^y.4)   5lM.7f   S^    9^ 

10    11  Y\^ 

2      4       5       6  j    7|    8       9  j  10     11 

12     13 

14% 

3       5       6       7  j    8  I    9  {  10     11     12 

IS     14 

15  i 

4       6       7  1    8  1    9  j  10     11     12     13 

14     15 

16 

'5       7       8       9  1  10  i  11     12     13     14 

15     16 

17"^ 

ij  ^{     8       9.     10  1  11  !  12     13     14     15 

16     17 

18^ 

i   7  i    9     10     11  I  12  t  13     14     15     16 

17     18 

19f 

'^  S  1  10     11     12     Ijji4|i5     16     17     18  1  19 

201 

|9jii      12     33     14  j  i5  1  IG     17     18     19  }  20 

21 J 

'10  f  12     13     14     15  1  16  j  17 J  18     19     20  (  21 

22 

Kl'-L  Tl  V I .  J  C  A  T  ID  N     T  ABLE* 


"I'l    2  i    S  i    4  :    5  i    6  I    7  i     8  1      9 

10|  11    Igi 

.2!    4!     6i 

8  i  10  i  12  i  14  1  16  i     18 

20i  22   24 

f  S\    6  i     9  1 

12  1  15  I  18  I  21  ;  24  i     27 

S0|  33    36 

4  1     8  1  12  1 

16  1  20  i  24  ;  28  1  S2  J    S6 

40|  44    48' 

d  i  10  !  15 

20  1  25  1  SO  1  35  1  40  i    45 

50   55    60 

i  6  j  124.  18 

24  i  SO  I  S6  \  42  i  48  |    54 

60    66   7^ 

W\  14)  21 

28  i  S5  i  42  5  49  i  56  \    63 

701  77   S\ 

*  8  i  16  1  24 

1  32  j  40  1  48  1  5d  ]  64  j    72  5    SOj  8S|  96 

9  i  18  I  27 

36  1  45  1  54  i  63  t  72  |    81  t    90]  99|108 

10  i  20  1  SO 

40  i  50  1  60  i  70  i  80  5    90  J  100J1101126 

U  1  22  i  33 

44  1  55  i  66  1  77  1  88  i    99  1  110|i21|132 

12  (  24  1  36  ; 

48  1  60  i  72  J  84  ;  96  !  108  !  1201132114^ 

To  learn  tins  Table  :  Eind  yam*  mijitipiier  in  the  left 
Irand  column,  and  tlie  mnltiplicaiul  a-top,  and  in  the 
conunon  angle  of  meeting,  or  against  your  multiplier, 
aloag  at  the  right  hand,  and  under  yoUr  fnultipIicJOidj,^ 
tt^j  vill  fiT^jJ  the  |irjT(IiUt^  or  ans^ver.. 


0 

ounces,                   1  pound;  lb  * 

.^  ..:.  >-s. .,/  .»U'ikb            1  ounce,  c-^. 

,  16  ounct5^,                            1  pound,  lb, 

I  ^8  poimdi?,  i  qifarf^r  of  a  hundred  ueigu^  r'n 

I  4  quarttn-^j                          1  liundred^velglitj  cjz'L 

1 20  hundiiid  w^iglif  •               1  ton,             -^  jtX 

\     By  iWs  weight  arc  vclghed  all  coarse  and  drcsSy  gpod:/. 
Igi-ocery  wares,  and  all  metals  except  gold  and  silver. 

4,  nSpofhecuHe^  Wei^d\ 

20  giralrra  (^\)  m&!;fc           1  scp«|iTe)  3 

5  scruples,                            1  dramt,  5_^ 

S  dranis,                                1  ounce^  5 

'TS  oanceK,                             1  pound,  f^ 

Apotliecavies  use  mis  wei^lit  In  CDrnpOundl-ng  thcTr 
medirines. 

5.  6?fcfi  Measure,  . 

4  nails  (7Z/r.)  niak'c                 1  quarter  of  a  j:u-d,  (p\ 

4  quart eirs,  '                         1  jan!,  ii<}. 

S -uartcr?^                           1  fell  FlemuHi,  KfL 

'    ;cart6rs,                             1  Ell  En^5isl'.,  j^.  )ih 

C  garters,                            1  Ell  French,         .  ii'J^Jf. 


t" 


■jjf.)  ina'.<re  T  qi^ro't,  o 


8  qjisr  r'^  I  peck; 

4  pecks.  I  bushel, 

'fhis  meaiiurei:^  applied  te,^r-»'^!:.  lenn^" 
ir.ts,  OTSters.  con'.  ^:o 


?. 


/ 

» 

I          -^ 

ARiTHi^IETIGAL  TABLES, 

^' . 

JVim  Measure, 

4  gills  {gi.) 

make 

1  pint. 

2  pints. 

1  quart, 

4  quarts. 

1  gallon, 

5.1  i  gallons, 

1  barrel, 

42  gallons, 

1  tierce, 

63  gallons, 

I  hogshead, 

2  hogsheads. 

1  pipe, 

2  pipes, 

1  tun, 

12 


qt. 

t 

tier, 
hhd. 

V' 
1\ 

All  brandies,  spirits,  mead,  vinegar,  oil,  S:c.  are  mea 
sured  bj  wine  measure.  Jfote,-^9>^\  solid  inches,  mak 
a  gallon. 


8.  Long  Measure. 

5  barley  coras  (&.  c.)  make  1  inch,  marked  in, 

J  2  inches,  1  foot,  ft, 

3  ftetj  1  yard,  yd. 

5 i  yards,  1  rod,  pole,  or  perch,  ri^. 

40  rods,  1  furlong,  fiu\ 

8  furlongs,  1  mile,  ?n. 

5  miles,  1  league,  ha, 

69^  statute  miles,  1  dep-ee,  on  the  earth. 

360  degrees,  the  circumference  of  the  earth. 

The  use  of  long  measure  is  to  measure  the  dlstonce  o 
places,  or  any  other  thing,  where  length  is  considered 
without  regai'd  to  breadth. 

N.  B.  In  measuring  the  heightof  horses,  4  inches  mak( 
1  hand.  In  measuring  depths,  six  feet  make  1  fatliopi 
or  Fsench  toise.  Distances  are  measured  by  a  chain 
four  rods  long,  containing  one  hundred  links* 


^ 


AHITUMETIGIAL  TABLES.  iS 

9.  Land  J  or  S^juare  Measure. 

144  square  inches  mak6  1  square  foat. 

9  square  feet,  1  square  yard. 

SOi  square  yarde,  or  >  i  square  red. 
272i  square  teet^       3 

40  square  rods,  1  square  rood. 

4  square  roods,  1  square  acre. 

640  square  acres,  1  square  mile. 

10.  Solid  or  Cubic  Measure, 

irSS  solid  inches  make  1  sclld  toot 

40  feet  of  round  timber,  or  >      .  .  ^      > 

50  feet  or  hewn  timber,        3 
1-28  solid  feet  or  8  feet  long,  >      j^,,,.^  ^j-  ^^^^_ 
4  wide,  and  4  nigh,  5 

All  solids,  or  things  that  have  len|^tb,  br  eadtli  and  deptii, 
are  measured  by  this  measure.  N.  B.  The  wine  gallon 
contains  251  solid  or  cubic  inches,  and  the  beer  gallon, 
282.     A  bushel  contains  2150,4£  solid  inches. 


60  seconds  {S.)  make 

1  minuie, 

mi 

irked- 

S.Mi 

€0  minutcsj 

1  hour, 

h. 

24  hours, 

I  day, 

(L 

7  days, 

1  week, 

u\ 

4  weeks. 

I  month, 

VIO 

13  months,  1  day 

and  6 

liours,  1  Ji 

Lilian  ycarj 

yr. 

Thnty  days  hath  September,  April,  Jane, and  Novembei , 
February  twenty -Kiiglit  alone,  all  the  rest  have  ihirty-one. 
N.  B.  In  bissextile,  or  leap  yeai*,  February  hath  29 
♦Jays. 

12.  circular  Motion* 

€0  seconds  ('')  make  1  minute,  ^ 

60  minutes,  1  degree, 

SO  degrees,  ;  sign,  S. 

*t  5?\;^iS;  or  560  degicei?,  ihe  ^,vho!c  great  Circus  cf  thf 


Ex^amlion  of  Chwracters  used  in  tkU  S^ok 


:^.  Kqiial  to,  as  l£i.  =  Is.  sigQliies  tFiat  i£  pence  a;c 
equal  to  1  shilling. 


-T  .*i^^ors,  iho.  sign  of  addition, 
that  5  and  7  added  toii-ethcr; 


-  -  Mr^nus,  or  /^ss,  tlie  s';. 
sigiiiiies  Hiat  2  subtrac:.;,^.  .. 


X  Multiply^  or  Yc'ilu?^  tlic  s:;^'--  ^-  M^''n-;'iciition ;  .^ 
4x3=125  signifies  that  4  :::!:. t'::  .  is  ecual  Uj 

12. 

-r-  The  sign  of  Division ;  cs  o^^~2-~^..  slgniiles  i}v^i  S 
d^ded  by  2;  is  equal  to  4  :  or  thus,  v~49  each  ^^r 
wliich  signify  the  sarne  thiDg*. 

:  :  Four  points  set  iu  the  muklieof  loi;i  niiuibers.  uc...>.. 
ilieni  to  be  proportional  to  one  another,  by  iwQ  rule- 
of  threes  as'2:4  :  :  8  :  16;  that  is/  as  B  to" 4*  se  is  o 
to  16, 

V*  Freilxed  to  anjni:ij:bc:>  -^upDCSCs  t^^at  lhcsqv:ar-^ 
of  thatiuumbcr  is  reotilreiL 

^'  Prefixed  to  any  ncHnbe:'-  •.■.{u^.  ^ir-j.--.. 

number  is  reouireu. 


v'  Denotes  the  bi^uadraie  root,  (>r  fourth  poxvcj^.  ''< 


ARITHMETIC. 

.  X-RITHMETIC  is  tlie  art  of  coinputiKg  by  riumbws, 
nnd  has  five  principal  rules  for  its  operation,  viz.  Numc- 
r.ilioi);  Addition,  Subtraction,  Multiplication,  and  Divf.- 


KUMERxVTIO?!. 

"'   ineriitioii  is  the  art  of  numbering.    It  teaches  to 

;s  the  value  of  any  proposed  number  by  the  follow- 

iiractcrs,  or  figures : 

1,  2,  5,  4,  5,  G5  r,  S.  9,  0— or  cypher. 

■".••-ides  tiic  simple  value  of  figures,  each  has  a  local 

due,  whicli  depends  upon  the  place  it  stands  in,  viz. 

^y  r?.i!;ure  in  the  place  of  units,  represents  only  its  sim- 

3  value,  or  so  many  ones,  but  in  the  second  place,  oi' 

Note. — Although  a  cypher  standing  alone  signifies  notli'- 
jjig  ;  yet  when  it  is  placed  on  the  right  hand  of  figures,  it  in-  ■'. 
crertscs  their  vaUie  in  a  tenfold  proporti'>n,  by  throwing  them  ^ 
•::tt>  higher  places.     Thus  2  with  a  cypher  annexed  to  il^ 

como«5  20,  twenty,  and  with  two  cyj^hcrs,  thus,  200,  two 

.  :!!K:rcd, 

fi.  When  numbers  consisting  of  many  figures,  are  given  to 
'  i  rcadj  it  will  be  found  convenient  to  divide  them  into  as 
any  periods  as  we  can,  of  six  figures  each,  reckoning  from 
0  right  hand  towards  the  left,  calling  the  first  the  period  of 
it:;,  (lie  second  that  of  million?,  the  third  billions,  the  fouith 
iHti^s,  &CC.  as  iii  the  followin;!;  nnm])er : 
4i0  7i3C£5  46  27  G  90  12  5  0  0  792 
Fcj-lod  of  \  S.   Period  of  |  £.    Period  of     J .    Penod  {f 


'^\!Ulons,      i       Billio'iis.  Millions 


Units. 
506792 


v.C7:>  I         625182         }         739012 

Till;  foregoing  number  is  read  thus— Eight  thousand  and 
v'  •ty-tlivee  trillions  ;   six  hundred  and  twenty-five  thou- 
'  >iTr  liitodred  and  sixty-iwo  billions  ;  seven  hundred  and 
•ine  tho'4r>r.nd  and  twelve  millions:  five  hundred  and 
r-ar.d,  seven  hundrt^d  and  njnety-two. 
:.  Billions  h  substitute  1  for  millions  of  miilions. 
'     iiions  for  millions  of  mr-Iions  ef  milliens. 
Quatrillions  for  miilionfj  of  milUi  ns  of  millions  of  mnHonsv 


phicQ  ol  tens,  it  becomes  so  many  tens,  or  ten  times  its 
simple  value,  aiul  in  t!ie  third  place,  or  place  of  hundreds, 
it  becomes  an  liii rKhed  limes  its  simple  value,  and  so  on, 

as  in  the  rblio^vi!: ; 


n  y,  g  p  X  ^  ^^  js'  C 

C  o  ^'  c  c  o  5  J3  g: 

^  hTI     I        Jj      £i     ^    ^ 


TABLE : 


Lc    o: 


1   .(h!(-. 
'   2  i   -  Twenty-one. 
.        •    •    '    "   S  2  1    -  Three  hundred  twentj-one. 
'    ,    ,    '4321.-  Four  thousand  321. 

5  4  3  2  1  -Fifty -four thousand  321. 
i    ,    '    6  5  4  3  2  1   -  654'thousand  321. 
,       7  6  5  4  3  2  1   -  7  miiiion  654  thousand  521. 

87654321   -87  million  654  thousand  321. 
987654321   -987  million  654  thousand  321. 
123  4  56789  -123  million  456  thousand  789 
9  8  7  6  5  4  3  4  8  -  987  million  654  thousand  348. 

To  knovr  tiie  value  of  any  number  of  figures. 

RULE. 

1.  Numerate  from  the  right  to  the  left  hand;  c  .ch  Sg- 
tire  in  its  proper  place,  by  saying,  units,  tens,  hundreds,  ^ 
ike,  as  in  tlje  Numeration  Table. 

2.  To  the  simple  value  of  each  figure,  join  tlie  name  of 
its  place,  beginning  at  the  left  hand,  and  reading  to  the 
right. 

EXAMPLES. 

Head  the.  ftdlowiiig  numbers, 

365,  Three  liunilred  and  sixty-live. 
5461,  ¥ive  thousand  four  hundred  and  sixty -one. 
1234,  One  thousand  tw^o  hundred  and  thirty -four» 
540^56,  Fifty-foirv  thousand  nnd  twenty-six. 


.......  Une liiintlred  and  twentj  .,..:.  liousind  four 

liundrcd  and  sixly-one. 

'if'G.':^40,  Four  millions,  six  hundred  and  six<y-si\'  (hou- 
- -nd  two  hundred  and  forty.  , 

I 

i.. — i.\n   convenience  hi  reading  lar^e  numbers, 
lay  be  divided  into  periods  of  three  figures  each, 

OVv\S  J 

OSr,  Nine  hundred  and  eighty-seven. 
VST  OOO5  Nine  hundred  and  ei;^;hty-seven  thousand. 
r  000  000,  Nine  huudred  and  eighty-seven  miHion. 
"  r,  !  f.'!*!.  Nine  luinch'ed  ;ind   eigtity -seven  million, 
six  hundred   and    iifty-four  thousand, 


Begin  on  tl.c  ri^j.l:  I'.aiid,  write  units  in  the  uni^ts  place, 

'  n.s  in  the  tens  place,  hundreds  hi  the  hundreds  ]?lace, 

d  so  on,  towards  the  left  hand,  vm ting  each  figure  ac- 

./iling  to  its  proper  value  in  numeration;  taking  care 

r:.upp]y  those  places  of  the  natural  erder  with  cyphers 

■tich  are  omitted  in  the  question. 

Yv'rite  down  in  ]^roner  ilscures  the  following  numbers 
Thirty-six.  " 

Two  hundred  tMid  s?eventy-nine. 
Thirty -seven  thousaiid,  five  Imndred  and  fourteen, 
-'sine  millions,  seventy -two  thousand  and  two  hundred. 
lit  hundredMn*;"*  "    ly-ibur  thousand  and  nftv- 


':,-i^- 


SIMPLE  AUDITION, 

'ilng  tor^ctlier  scveiiil  smaller  n?j!nber??,  of  iht^ 
!''tominatioti,  into  one  larger,  equal  to  the  whole. 
'.  ti;tal ;  as  4(h>Ilarsniul  six  dollars  in  one  sum  is  10 


13  SIMPr.K   AJlDVrtON^ 

UULE. 

Having  placed  units  under  units,  tens  under  tens,  &c. 
draw  a  line  underneath,  and  begin  with  the  units ;  after 
adding  up  every  figure  in  that  column,  consider  how  ma- 
ny tens  are  contained  in  their  sum  ;  set  down  the  remain- 
der under  the  units,  an^^  carry  so  many  as  you  have  tens, 
to  the  next  column  of  tens ;  proceed  in  the  same  manner 
tTiroug]]  every  column,  or  row,  and  set  down  tlie  whole 
amount  of  tlie  last  row. 

JiiXAMPLKS. 

(!.)  (2.)  ,S.)  (4.) 


Hundreds. 

Tens. 

Units. 

Tliousands. 
Hundreds- 
Tens. 
Units. 

C.  of  Thous: 

X.  of  Thoiisj 

Thousands. 

Hundreds. 

Tens. 

Units. 

4  2 

4  1  4 

17  5  6 

5  5 

2  6  2  1 

5   3 

2  9  1 

0  4  3  2 

3  4  6  9  7  7 

5  2 

8  5  1 

9  4  7  8 

4  1 

3  3  3  9 

1  3 

i  5  2 

16  6  6 

3  2 

10  12 

8  9 

6  9  8 

7  4  2  2 

8  7 

6  5  4  5 

(5.) 
5  14  8  5 

(6.) 
64  179 

S  7  1  4  5 

6  7  2  3 

7 

2  5  7  12 

5  17  14 

4  2  7  1 

9 

8  4  19  4 

6  0  8  4  5 

9  7  14 

5 

S  2  5  1  6 

5  7  3  5  7 

S  2  8  5 

1 

7  1  4  S  2 

6  17  8  4 

14  5  7 

2 

S  27  1  9 

5  2  10  1 

* 

r>  1 


siMj*:. 

F.  a;) in  J 101 

19 

...,; 

(9.) 

(to.) 

8  4  rs 

8 

5  2  6  3  7 

C-.      'V      O 

9 

3  7  I 

4 

2  7  19  6 

2  5  i) 

•3714 

" 

3  8  4  19 

4  1  7 

1 

8  3  2 

1 

5  3  19  2 

7  2  r> 

t 

1  4  3 

/ 

6  10  8  4 

5 

I  r  2 

6 

3  7  19  5 

/^ 

2  5  1 

b 

2  9  14  7 

• 

• 

(12.) 

V'  4  yt  3  :  r  8 

5!  9 

3  r 

1 

8  4  5  6  8  7 

r  4  2  T   0  6  1 

f)  a 

5  1 

1 

7  0  4  2  2  9 

r  1  0  0  4  r.  r 

9  6 

I 

9  4  6  0  3  7  2 

r  G  2  3>  1  4  5 

"r   c*» 

8 

3  4  0  7  5  4 

(^  0  (•  0  4  1   B 

S  4 

2  7  0  15  5 

7  0  4  13  6  0 

5  3 

3  6  0  2  3 

;■  6  r  8  0  9  S 

p  ^ 

,__. 

_ 

19  5  0 

(15.)    ' 

(14.) 

:^  4  3  0  6  4  G 

2  5  9  0  0 

f)  B  8  14  5  1 

3  4  0  0  4  5 

•116  0  4  5  2 

5  4  0  4  4  3  3 

7  6  10  4  2  5 

3  7  0  5  5  3  2  6 

3  4  G  2  1  4 

4  0  5  2  17  4 

4  0  3  0  9 

4 

06476269 

9  8  2  r 

2  0  6  8  5  9  1 

j^To  prove  Addition^  ue^pii  at  the  fep  of  the  sum, 

reckon  the  figures  dovvnwardsin  the  same  manner  as 

'  were  added  upwards,  and  if  it  be  right,  this  sum  total 

.  bo  equal  to  the  Hrst:   Or  cut  off  the  upper  line  of 

:^  !  ■'«,  and  find  t\\Q  amount  of  the  rest;  then  if  tlie  amount 

j^;w'?  lipper  line,  when  added.. be  ef«^i>al  to  the  total.  {\^^ 

I  ^\iKvk  U  sf?Tino?.ad  to  be  rid  it. 


r> 


2.  There  is  ruu^Lhirr  ;ueL:io(l.  oT  prooi^  as  ibiiows  : — ■ 

Reject  or  cast  out  l:;e  nines  in  each      example, 
vow  or  ^v.i^u  of  Tigures.  ajid  set  down  thQ     S  7  8  2 
'^n.K  i  ideis,  eacii  directly  even  with  the     5  7  6  6 
ligurcs  In  itsniv/:  find  the  suin  of  these     8  7  5  5 

reniraiiders  ;  then  if  the  excess  of  nines      » 

in  t\ui  sum  found  as  before,  is  equal  to  the   18  vS  0  3 

evcess  of  nines  in  the  sum  total^  the  work 

is  siioposed  to  be  rii^ht. 

•*15*  Add  86S5,   ^194^  7421,   5063,   2196,   and    1245 
f  together  *Rns,  26754. 

16.  Find  the  sum  of  S4S.%  783645,  318,  7530,  and 
9078045.  '  ^ns,  ir)473020. 

IT,  Find  the  ':uni  total  of  604,  4680,  98,  64,  and  54. 
*ins,  Fi  rt  V  -fiYO  UHndred. 

13.  VHiat  is  the  sum   total   of  24674,  16742,  34678, 
10467,  and  1 3459  ?  ^Ins.  One  liiin<lred  tliousand. 

19.  Add  1021,  24S9,  28763,  289,  a!id64S8  together. 

dns.  Forty  diousand. 


^0. 


\\  na.1: 


:Ni  lie  1^)11 


::in  total  of  the  following  numbers,  viyrj 
id 4005  ?  Jm,  UUi. 

i;m  total  of  the  following  numbers,  vi?:. 

and  forty-seven, 

1  six  Ihundred  and  EvCj 

sand  si"  hTnidrec!. 


^Hc  ana  ^ 
s.  and  t 

Hions,  a 


ne  liiou^anU  , 


c^;i^;;ctr,  61374177 


g2.  Rcq  :  sum  of  the  following  numbers,  vi*?:. 

Five  'imuuvi'ii  and  sixty-eight. 
Eight  thousand  eight  hiidd red  and  five, 
Fkiventj-nlne  thousand  six  hundred, 
Nine  hundred  and  eleven  thousand, 
Kine  milhons  and  tv/cmtv-bi^v 


Ansmr^    9999D')^ 


qUESTIONS. 

t    Wnai  number  oi  dollars  are  in  sii:  bagSj  containing 

ench.  37542  dollars  ?  ^^ns.  2So252,    • 

".  If  one  quarter  of  a  ship's  cargo  be  worth  eleven 

;sand  and  ninctj-nino  dollars,  how  many  doUai-s  Is  the 

Ic  cargo  worth  ? 

Ans.  44S96dols. 
;  .  Money  was  first  made  of  gold  and  silver  at  Argos, 
o'-}\i  hundred  and  ninety-four  years  before  Christ;  how 
:;  has  money  been  in  nse  at  this  date^  1814  ? 

Jhis,  2708  years. 

.  Tlie  distance  from  Portland  in  the    Province  of 

•lie.  to  Boston,  is  125  miles;  from  Boston  to  Isev/- 

eu,  162  miles;  from  lUence  to  New-York,  88  ;  from 

ICO  to  j^liilatleiphia,  95;  from  thence  to  Baltimore, 

iw.l;  from  thence  to  Charleston,  South-Carolina,  716; 

rv.d  fro^a  thence  to  Savannah,  119  miles — What  is  fliQ 

V.  !-<ie  distance  from  Portland  to  vSavannah  ? 

diis,  1407  miles. 
5.  John,  Thomas,  and  Harry,  after  counting  their 
pnxe  money,  John  had  one  thousand  three  hundred  and 
fiCventy-five  dollars  :  Thomas  had  just  three  times  as  ma* 
ny  as  Jolm  ;  and  Harry  had  just  as  many  as  John  and 
'1  iiomas  both — Pray  how  many  dollars  had  Harry  ? 

Jns,  5500  dollars. 

FEDERAL  MONEY. 

-^  EXT  in  point  of  simplicity,  and  the  nearest  allied  to 
:1c  numbers,  is  the  coin  of  the  United  States,  or 

FEDERAL  MONEY, 

.'his  is  the  most  simple  and  easy  of  all  money — itln- 

;  cs  in  a  tenfold  proportion,  lilie  whole  numbers. 

10  mills,  (7«.)  make,     1  cent,  marked   c. 

iO  cents,  1  dime,  d, 

JO  dimes,  1  dollar,  S. 

10  dollars,  1  Eagle,  ^  E, 

)olIar  is  the  money  unit;  all  otlier  denominations  b05 

revalued  according  to  their  place  from  the  dollar's 

place.    A  point  or  comma,  called  a  separairix^  iT\oy  be 

r.*>-T^T»i'  nfrr^r;.  ;>>^  f>rt!rMrs  to  separate  fh^in  from  ftQlli&apir 


ci^noiDUiations ;  thjeu  the  first  figure  at  the  ilght  of  this 
separatr]x  :s  dimes,  the  second  figure  cents,  and  the  third 
mills.* 

^^.^ /'^ 

ADDITION  OF  FEDERAL  MONEY.       "'^'' 
RULE. 

2.  Place  the  numbers  accoi-diiiff  to  their  value ;  that  is, 
'3lkrs  under  dollars,  dimes  under  dimes,  cents  under 
.-its,  &c.  and  proceed  exactly  as  in  whole  numbers; 

^n  place  the  separatrix  in  the  sum  total,  directhMindc!? 
.e  separating  points  above. 


KXAMPLKS. 

s. 

d.  c.  w. 

S.    d.  cm. 

S. 

d,  c.  m\ 

sas. 

5  4  1 

439,     5  0  4 

136, 

5  14 

487', 

0  6  0 

416,     3  9  0 

125, 

0  9  0 

94, 

6  7  0 

168,     9  3  4 

200, 

9  0  9 

439, 

0  8  9 

539,     0  6  0 

304, 

0  0   6 

74-2, 

5  0  0 

143,    0  0  5 

111, 

19  1 

2128,     8  6  0 


2.  When  accounts  are  kept  in  dollars  and  cents,  and 
no  othei  dienominations  are  mentioned,  which  is  the  usu- 
al mode  in  common  reckoning,  Uien  the  two  first  figure^ 
at  the  right  of  the  separatrix  or  point,  maj  be  called  so 
many  cents  instead  of  dimes  and  cents  ;  for  the  place  of 
dimes  is  only  the  ten's  place  in  cents ;  because  ten  cents 
make  a  dime ;  for  example,  48,  75^  forty-eight  dollai'S, 
seven  dimes  five  cents,  may  be  read  forty-eight  dollars 
and  seventy -five  cents. 

^  It  may  be  observed  that  ail  the  figures  at  the  left  hand 
of  the  separatrix  are  dollars  ;  or  you  may  call  the  first  figure 
dollars,  and  the  other  eagles,  he.  Thus  any  sum  of  tliis 
money  may  be  read  differently,  either  wholly  in  the  lowest 
denomination,  or  partly  in  the  higher,  and  partly  in  the  low- 
est ;  for  example,  S7  54,  may  be  either  read  S754  cents,  oT 
S75  dimes  and  4  cents,  or  37  dollars  5  d/mes  and  4  ccnt^,  oV 
S  eagles,  7  dollars  ^'  dimes  and  4  centa# 


A 


Ifiiie  cents  are  less  than  ten,  place  a  cypher  in  flie 
ffen's  place,  or  place  of  dimes. — Example.  Write  do^v^l 
four  dollars  and  T  cents.    Thus,  84,  07  cts, 

EXAMPLES, 

1.  Find  tlie  sum  of  504dollai*s,  39  cents;  291  dollars, 

9  cents;  136  dollars,  99  cents;  12 dollars  and  lOcfrnts^ 

r304,  59 

,p,  J  291,  09 

^^^^'       •    156;  99 

I  \%  10 

Sum,    744^  57    Sevon  hundred  forty-four  dol' 
— lars  and  fiftrj^-scvcn  cents* 


(2.) 

(S.) 

W 

S.  c/5.- 

S.  c's> 

g.  cts^ 

0,  99 

364,  00 

3287,  80 

0-  50 

21,  50 

1729,  19 

0,  25 

8,  09 

4249,  99 

0-  75 

0,  99 

140,  01 

[0.) 

(6.) 

(7.) 

». 

S.   €ik?. 

g.  cf^. 

£468 

124,  50 

,  165 

1900 

9,  07 

,  99 

24G 

0,  60 

,  861 

H5 

£31,  01 

'  'L 

167 

0,  75 

^    Q7ti 

46 

24,  00 

,     72 

•]  \i 

9,  44 

,  99 

o 

0,  95 

,  09 

..  Wi.at  IS  tnesiuu  total  of  127  dols.  19  cents,  278 
doli    19  cents,  5.4  dols.  7  ceuts^  5  dols.  10  cents,  and  J 


9.  Wliirt  is  ibeBuiii  (U  S7"'8  dois.  1  cl,  ..:.,  .iC,....  _  . 
S44  dolso  8  cts.  and  S63  ('"oi.^.  ?  ./??z<-.  gl^-^- 

10.  Wliat  is  the  sum  Oi  4G  cents.  i;C  cciiis,  G;2  c:.; 
and  10  cents  ?  '  J??i;,  S^- 

■  i  I .  What  is  tue  bUm  of  9  dlaies^  S  dimes,  ViVaI  SO  •;  :■ :  : 

'1^.  I  received  «>!'  A  E  and  C  a  su:.i  c<:'  ;uu:-  ^ 
paid  me  95  dols.  43  ci^.  B  paid  me  juso  three  ■■:  : 
much  as  A,  :uid  G  "paid  mc  jii^-t  as  much  as  .'.    r:  •.. 

bnth  5  ciiiiy^:i  ;:.d!  me  lunvm'  •       ,  .    ..  -  .    ^  .   ;    ^     •. 


838  cents;  and  o;:- •  -..:■ ''   her  c^i^)  iBv^o^h:  > 

dok.  65  csnt^js     '-';,. y  whilst  m  the  valiie  of*  the  wi>ele  i,:..> 
and  curgor  "  ,/ijti.s'.  81137^2,  4dcts.  ' 


A  TAILORS  BILL. 

Jlfr.  Janies  Taifwcll^ 

To   Timolhy  Taijlui'^  ]J/, 

April  15.      To  2^  yds.  uf  Cloth,  at  6,  50  p^r  yd.  16  ;:  > 

To  4  yds.  Shalloon,           75         "  S  ■:.:> 

To  making  your  Coat^  2.  J 

To  1  silk  Vest  Piittern,        f  4  IJ 

To  making  your  Vest,  1  50 

To  Silk,  iButtonSj  &c.  for  Veiit,  0  4J 

Slim,  S  ^i^  -^^^ 


li*^  By  au  act  ot  Congress,  all  tlie  accaiints  of  t'^ij 
United  States^  the  salaries  of  all  officers,  the  reveiii:  .^ 
&c.  ai^e  to  be  reckoned  in  federal  money :  which  made  *;  c 
reckoning  is  so  simple,  easy  and  c^^  v*^.^!/ v,;.  :% ,.  .  v.  ; 
fcc on  ct}mc  in :d  common  practice  ir 


SIMPLE   SUBTllACTiON.  i':? 

SIMPLE  SUBTRACTION. 

Subtraction  of  whole  J^'^umba^) 

1  EACHETH  to  take  a  less  number  from  a  gi-eiiter,  qI 
the  same  denomination,  and  therebj  shows  the  difference^ 
or  remainder  :  as  4  dollars  subtracted  from  6  dollars,  th« 
remainder  is  two  dollars. 

RULE. 

Place  the  least  nuiixbcr  under  the  ;:^reatcsr.  so  thai  units 
may  stand  under  units,  tens  under  tens,  &c.  and  draw  a 
line  under  tliem. 

S.  Begin  at  the  right  hand,  and  take  each  figui'e  in  the 
Imver  lipe  fitnn  the  iigure  above  it,  and  set  down  fhe  re* 
mainder. 

S.  If  the  Ixiwer  figure  is  greater  flian  fliat  obova  it, 
add  ten  to  the  upper  iigure  ;  from  whicli  number  so  in- 
creased, take  the  lower  and  set  down  the  remain/iei*,  car- 
rjing  one  to  the  next  lower  number,  with  \yliich  proceed 
e§  before,  and  so  on  till  tlie  v/hole  is  finished. 

mooF.  Add  the  remainder  to  the  least  numbt^r,  and 
if  tlie  sum  be  cqtial  to  the  greatest,  the  work  is  r]ght 


QreaUsimmher,Q^4  6  8      G  2.\ '5  T 
Leastmmber^      1346      12148 

8  7  9^6  4  7  5 
16  4  3  4  8  9 

Difterenccj 

i^root,                                 ■ 

(4.)                       f5.) 
From    416^8839         9187'64ofK) 
Take     S1542999           9124'S8a6 

(5.) 
65432167890' 
1 23456?; 098 

Rra 


S6 


^;lMrL>:   Sl/BIRAOIIOX. 


Take 


'"'17144045605 
40600832184 


5562176255002 
1235271082165 


Him. 

(9.) 
From    i  00000 
Take      65321 


Bift: 


2521665 
2000000 


(11.) 
200000 
99999 


(12.) 
10000 


Ms.  66666. 
dnn.  730865. 
..3/Z.s.  142444. 

Ms.  90037. 
..^?2S.  250822. 


13.  From  560418,  take  293752. 
l^.  From  765410,  take  34747. 

15.  From  341209,  take  1987G5. 

16.  From  100046,  take  10009. 

17.  From  2657804,  take  2376982. 
IS.  From  ninety  thousand,  fivo  hundred  and  forty-^iX^ 

take  forty-two  thousand,  oae  hundred  and  nine. 

JhTS.  4843r. 
r9.  From  fifty-l^irr  Qiousand  and  tuiinty-six,  talce  itSflo 
tiiousand  two  hundred  and  fifty -four.         /nis.  44772. 

20.  From  one  million,  take  nine  hundred  and  ninety- 
nine  thousand.  Ans.  One  thousand. 

21.  From  nine  hundred  and  eighty-seven  millions) 
talce  nino  hundred  aiid  eighty-seven  thousand. 

Jxns.  986013000. 

22.  Subtract  o^e  from  a  million,  and  shew  the  remaih- 
ocr.  '  Ms.  999999. 

qUESTIOSS. 

! .  Hcv;  inudi  is  six  hundred  and  sixiy-seven,  greater 
J;an  three  hundred  and  ninety -live  ?      *       Arts.  272. 

2w  What  is  the  dilTsrencebetvveen  tvv'ice  tv/cnty-seven, 
a  ad  €iree  times  forty -five  ?  Ms.  81. 

S.  How  nmich  is  1200  greater  than  565  ami  721  added 
togetlier?  ^         Ms,  114. 

4.  From  Kew-London  to  Pluladelphia  is  240  miles. 
Now  it'  a  man  should  travel  five  days  from  New-London 
towards  rhiladelphia J  at  the  rate  of  39  mil23  each  dav, 
'GW  far  wuXil^  he  tbei:  be  from  Philadelphia. 

M^,  AS  rxii\\iF' 


SIJBTIl ACTION   OF   JFi-OEUAI^   MQSlZY,  U7 

5i  What  other  number  with  these  four,  viz.  Gl,  32,  Viy 
and  12,  ^vill  niake  100  r  .pis,  19. 

6.  A  wine  merchant  bought  721  pipes  of  wine  for 
908-16  dollars,  and  sold  543  pipes  thereof  for  85049  dol- 
lars  ;  how  many  pipes  has  he  remaining  or  unsold,  and 
what  do  they  stand  him  in  ? 

JJtis,  178  pipes  unsold,  and  they  stand  him  in  31797. 


SDBTR ACTION  OF  FEDEKAT.   ]VIO>fEY. 


liULE. 
Place ihe  numbers  according  to  their  value;  ihat  :?, 
iTolIai©  under  dollars,  dimes  under  dimes,  cents  under 
cents,  &c.    and  subtract  as  in  whole  numbers. 

EXATVIPLKS 

g.  d,c.  in. 
From  45,  *  4  7  5* 
Talce  43j    4  8  5 


K(5m.  SI,    9  9  0  one  dollar,  nine  dimes,  and  nke  centSf 
or  one  dollar  and  ninety-nine  cents. 
g.     d,  c,  g.     d,c,vi,         g.     d,  c.vf, 

from    45,    7  4  46,    S  4  6        £il,     1  1  0 

Take    13,    8  9         36,    l  6  4        ill,    114 

Rem.  '        " 


g.  g.  cts. 

From   4  2  8  4  411,  24 

Take    19  9  3  16,  09 

Rem.    " 


g. 

960, 
13G, 

cts, 
GO 
41 

g.    cis,  g.    cts.  g.  cis. 

From   4106,' 71  1901,08  S 65,  GO 

Take     221,  69  864,  09  109,  01 


11.  From  l^doHarsj  take  9  dollars  9  cents. 

Jlmi,  gllj,  91  cf^. 

12.  ^ifim  127  dollars  1  ccr.t  take  41  dollars  10  cents. 

Ms,  885,  91  cts 


^•B  SXMPLK   MUl.Tj.PI.XaAXXON. 

15.  From  355  dollars  90  cents,  take  168  dols.  99  cents, 

Ans.  ^196,  91  cts. 

34.  From  249  dollars  45  cents,  take  180  dollars. 

dns,  S69,  45  cts. 

15.  From  100  dollars,  take  45  cts.    .^ws.  !g99,  55  cts. 

16.  From  ninety  dollars  and  ten  cents,  take  forty  dol- 
lars and  nineteen  cents.  Ans,  ^49,  91  cts. 

17.  From  fortj-one  dollars  eight  cents,  take  one  dollar 
nine  cents.  Ms,  ^39,  99  cts. 

18.  From  3  dols.  take  7  cts.  Ais,  !g2,  93  cts. 

19.  From  ninetj-nine  dollars,  take  ninety -nine  cents. 

.^ns,  S98,  1  ct. 
SO   From  twenty  dols.  take  twenty  cents  and  one  mill 
.^ns.  S19,  79  cts.  9  mills. 

21.  From  three  dollars,  take  one  hundred  and  ninetj- 
nine  cents.  '  Ans.  ^1^  1  ct 

22.  From  20  dols.  take  1  dime,      dns,  S19,  90  cts. 

^  23.  From,  nine  dollars  and  nioety  cents,  take  ninety- 
nine  dimes.  Ans.  0  remains. 

24.  Jack's  prize  money  was  219  dollars,  and  Thomas 
received  just  twice  as  much,  lacking  45  cents.  How 
much  money  did  ThomaiJ  receive  ?   Am,  g437,  55  cts. 

25.  Joe  Careless  received  prize  money  to  the  amount  of 
1000  dollars;  after  which  he  lays  out  411  dols.  41  cents 
for  a  span  of  fine  horses;  and  12-3  dollars  40  cents  for  a 
gold  watch  and  a  suit  of  new  clothes  5  besides  359  dol^ 
and  50  cents  he  lost  in  gambling.  Hoy/  much  will  !-ft 
nave  left  after  paying  his  landlord's  bill,  which  amounts 
to  85  dols.  and  11  cents  ?  Ans,  S20,  53  els. 

SIMPLE   MULTIPLICATION,  : 

X  EACIIETH  to  increase,  or  repeat  the  greater  of  two 
numbers  given,  as  often  as  there  are  units  in  the  less,  or 
multiplying  number;  hence  it  performs  the  work  of  ma- 
ny additions  in  tha  most  compendious  manner. 
*^The  number  to  be  multiplied  is  called  the  multiplicand. 

The  number  you  multiply  by,  is  called  themultipliei*. 

The  number  found  from  the  opemion,  is  called  tha 

p!OfllfCt> 


NoTK.     Botli  multiplier  and  multiplicand  ai «:  .a  ^vnc- 
ral  called  factors,  or  terms. 

CASE   L 
When  themuItlpliGr  is  not  mere  than  Iv/clve. 
RUI.E. 
Multiply  each  figure  in  tlie  nuiitiplicand  bv  tlie  multi- 
plier; carry  one  for  every  ten,  (as  in  addition  of  whole 
numbers)  and  you  will  have  the  product  or  answer. 
PIIOOF.* 
Multiply  the  multiplier  by  the  multiplicand, 

EXAMPLES. 

"Whatmumber  is  equal  to  3  tiincs  S65  ? 

Thus,    Z'j5  multiplicand, 
3  inidiinlier 


Ans,  1095  jiroiliicL 

90r5 

(i 


Midtiplicand 
Midtipller 

746 

)35 

i34 

a 

5432 
4 

S12GI 
9 

2345 
6 

Product 

4r0[)4 

4 

)46 

no 

4S£0 
10 

143£( 

2£406iS 
1:2 

4681114 

CASE  11. 

When  tlie  muUipiier  consists  of  several  iigiUTH. 

'      ilUI.E. 

The  multiplier  being  placed  under  th(^  niulliiJicand 

units  under  units,  tens  under  tens,  ^<c.  multiply  by  each 

significant  figure  in  the  multiplier  separately,  phicing  ihi: 

first  figure  in  each  product  exactly  under  its  multij>iier  ; 

*  Multiplication  may  also  ht*  proved  ity  ca^tin-g  out  iho  V/a 
in  Che  two  factors,  and  settin.:^  down  the  r^'iiiainders ;  then 
multiplying  the  two  remaindLTS  togetlier  ;  vi'  the  e.vcoss  u( 
9'3  in  their  product  is  equal  to  the  excess  of  \i'^  h\  the  tctal 
piDduct,  the  work  h  supposed  to  he  np.K> 


50 


fllMPLB  UVLTlVLXSiATlOiK* 


then  add  the  several  products  togBther  ia  the  same  ordej 
as  they  stand,  and  their  sum  will  be  the  total  product. 


EXAMPLES. 


What  number  is  efual  to  47  times  365  ? 

Multiplicand    3  6  5 
Mvltiplier  4  7 


1 

Ms.  1 

34293 
74 

9^5  5  0 
4  6  0 

7  15  5  product 

MSiltlpllcitmh     5rS64 
Midttfi'kY,            209 

4704^ 
91 

540776 

75728 

VvoducL.          7915576 

25S7682 

4280822 

8253       25203 
826        4025 

2193       9876 
4072       9405 

6816978    101442075 

8929896   92883780 

2G9181 
4629 

261986 
7638 

40634 
42068 

1246038849    2001049068 

1709391112 

154092 
87562 

918273645 
1003245 

11714545304 

921253442978025 

14.  Multinly  760483  by  9152.      .^is.  695994041G. 

15.  What  is  the  total  product  of  7608  times  365432  ? 

^ns.   2780206656. 

16.  What  number  is  equal  to  40003  times  4897685  ? 

Ms.   195922093055. 


SIMPLE   MUiLTirHCATlO.N. 


QASE  III. 
^Vhcn  there  are  cyphers  on  tlie  right  band  of  eitlier  or 
bnth  oF  tlic  factors,  nedect  those  cjimers ;  then  place  the 
significant  figures  under  one  another,  and  multiply  by 
tliem  only,  and  to  the  light  hand  of  the  product,  place  as 
many  cyphers  as  were  omitted  in  both  the  factors. 
examplp:s. 
21200  31800  84600 

70  36  34000 


(484000 


1144800 


55926000 
S040 

109215040000 


2876400000 

82530 
98260000 


U09397'800000 


7065000x8700=61465500000 

749643000  X  695000  ==»52100 1 885000000 

360000x1200000:^^432000000000 

CASE  IV. 

Wlmi  the  multiplier  is  a  composite  number,  xli at  s, 

when  it  is  produced  by  multiplying  any  two  nunioers  m 

the  table  together ;  multiply  fn-st  by  one  of  those  figur-es 

and  that  product  by  the  otlier;  and  the  last  prodiict  w^l 

be  the  total  required. 

EXAMPLES. 

Multiply  41364  by  35. 


289548  Product  of 


1447740  Product  of  5J 


SL  Multiply  764131  by  48. 

3.  Multiply  342516  by  56, 

4.  Multiply  209402  by  72. 

5.  Multiply  91738    by  81. 

6.  Multiply  54462    by  108. 
7   Mirltiply  615245  by  144. 


£ns.  36073288. 
.^iis.  19180896. 
aiis.  15076944. 

dns.  7430778. 

Jins.  3721896. 
.ins.  Si§59i99Sl. 


»»  .SIMPLE  MULTIPLICATION. 

CASE  \\ 
(To  multipl J  by  lo,  100,  1000,  &c.  annex  to  the  riniL 
%licji;id  all  the  cyphers  in  the  multiplier,  and  it  will 
OJiake  the  product  required. 

EXAMPLES. 

1.  Multiply  SS5    by  10.  Ms,  3650 

2.  Multiply  4657  by  100.  Ans.  465700 

3.  Multiply  5224  by  1000.  Ms,  5224000 
4    iVluitiply  26460  by  10000.  *i$?2S.  264600000 

EXAMPLES   FOIi    EXERCISE. 

1.  M^AirMj  1  £03450  by  9004.  Ms.  10835865800 

Go  Multi;!  -  90870G1  by  56708.  Ms,  51530905518S 

S.  Bfuitiply  8706544  by  67089.  Ms,  5841133S0416 

4.  Multiply  4321209  by  123409.  Ms,  533276081481 

5.  iMultiply  3456789  by  567090.  Ms,  1960310474010 
a  Multiply  8496427  by  874359.  Ms,  7428927415293 

98763542x98763542=9754237228385764 

Application  and  Use  of  Multijilication* 
In  making  out  bills  of  parcels,  and  in  finding  the  value 
of  goods ;  wrien  the  price  of  one  yard,  pound,  ike,  is  sp»v- 
en  (in  Federal  Money)  to  find  the  value  of  tlie  whole 
quantity. 

RULE. 
Multiply  the  given  price  and  quantity  together,  as  in 
whole  numbers,  and  the  separatrix  will  be  as  many  figures 
fcrom  the  right  hand  in  the  product,  as  in  the  given  price* 

EXAMPLES. 

1.  What  will  35  yards  of  broad-  >  g.  d,  c.  w. 
doth  come  to,  at   "  3  3,  4  9  6  per  yard  ? 

3  5 


17  4  8  0 
104  8  8 


Ms.  S122,  3  6  0=122  dol 
[lars,  S6  cents. 
^4  What  cost  35  lb.  cheese  at  8  cents  per  lb.  ? 
>08 

4to^.  ^^  80—2  dollars,  80  cctits. 


SJMELE   MULTirLlCATION.  53 

3.  WTiat  is  the  value  of  29  pairs  of  men'«  shoes,  at  1 
dollar  51  cents  per  pair?  ^ns,  g43,  79  cent*. 

4.  What  cost  131  yards  of  Irish  linen,  at  3&  ^ents  pc» 
yard  r  Jns,  g49,  78  c^nts. 

5.  What  cost  140  reams  of  paper,  at  2  dollars  3c«  tent:* 
per  ream  ?  Ms,  g,'»9. 

6.  What  cost  144  lb.  of  hyson  tea,  at  3  dollars  o  I  cenU 
per  lb.  .^  •Ans.  S505,  44  cents. 

7.  Wliatcost  94  bushels  of  eats,  at  33  cents  per  bush 
el  ^  Jins.  g31,  2  cents. 

8.  What  do  50  firkins  of  butter  come  to,  at  7  dollars 
14  cents  per  firkin  ?  *  Jins.  g357. 

9.  What  cost  12  cv/t.  of  Malaga  raisins,  at  7  dollaib 
SI  cents  per  cwt.  ?  Jlns,  g87,  72  cents. 

10.  Bought  37  horses  for  shipping,  at  52  dollars  pei 
head  ;  ^vhat  do  they  come  to  }  /ins.  gl924. 

11.  Wliat  is  the  amount  of  500  lbs.  of  hog's -lard,  at  15 
cents  per  lb.  ?  •Sns.  g75. 

12.  What  is  the  value  of  75  yards  of  i^atin,  at  3  dollarn 
75 cents j3er yard?  ^ins,  S281,  25  cents. 

13.  What  cost  367  acres  of  land,  at  14  doh.  67  cents 
per  acre  ?  Jlns,  g5583,  89  cents. 

14.  What  does  857  bis.  pork  come  to,  at  18  dols.  93 
cents  per  bl.  ?  Ms,  gl6223,  1  cent 

15.  What  does  15  tons  of  Hay  come  to,  at  20  dols.  78 
Cts*  per  ton  ?  Ms.  g311,  70  cents. 

16.  Find  the  amount  of  the  following 

BILL  OF  PARCELS. 

New-London,  Marcii  9,  1814. 
Mr,  James  Faywell^  Bought  of  William  Merchant* 

S^.  cts, 
28  lb.  of  Green  Tea,  at  2,  15  'p^v  lb. 

41  lb.  of  Coffee,  at  0,  21 

34  lb.  of  Loaf  Sugai-,  at  0,  19 

13  cwt.  of  jMalaga  Raisins,  at  7,  31  per  cwt. 
S5  firkins  of  Butter,  vX  7,  14  perjir. 

27  pairs  of  worsted  Hose,   at  1,  04  per  pair. 
^  buslicis  of  Oats,  at  0,  33  per  hush. 

£9  pairs  of  men's  Shoes,      at  1,  12  per  pair. 


Jlmount^  S5Jll>  78« 
I^cciyed  pavincnt  in  full,  W^jlliam  Mei?ojia;;t 


ClVI}5l6ji    or   WHOLE    KUMHEKS. 

A  SHORT  RULE.  

¥oTE.  The  -auieof  100  lbs,  of  any  article  will  be  jus! 

as  man  J  doiiars  as  de  article  is  cents  a  pound. 
For  1001b.  at  1  ceiUper  1  b. = 100  cents  =rl  dollar. 
1001b.  of  beef  at  4  :ents  a  lb,  comes  to  400  centS5=4l 

dollars,  §cc. 


DIVISION  OF  WHOLE  NUMBERS. 
Simple  division  teaches  to  find  how  many  limes 
one  whole  number  is  contained  in  another;   antl  also 
what  remains  ;  and  is  a  concise  way  of  performing  seve- 
ral subtractions. 

Four  principal  parts  are  to  be  noticed  in  Division  : 

i.  The  Dividend^  or  number  given  to  be  divided. 

9..  The  Divisor^  or  number  given  to  divide  by. 

S.  The  Quotient^  or  answer  to  the  question,  which 
shows  how  many  times  tlic  divisor  is  contained  in  ih^ 
dividend. 

4.  The  Remainder^  which  is  always  less  than  ih^  di- 
visor, and  of  the  same  name  with  the  Dividend. 
RULE. 

First,  seek  how  many  times  the  divisor  is  contained  in 
as  many  of  the  left  hand  figures  of  the  dividend  as  are 
just  necessary,  (that  is,  find  the  greatest  figure  that  the 
divisor  can  be  multiplied  by,  so  as  to  produce  a  product 
that  shall  not  exceed  the  part  of  the  dividend  used)  Vvhen 
found,  place  the  figure  in  the  quotient ;  multiply  the  di- 
ffisorby  this  quotient  figure;  place  t'he  product  under 
Aat  part  of  the  dividend  used ;  then  subtract  it  there- 
from,  and  bring  down  the  next  figure  of  the  dividend  to 
the  right  hand  of  tiie  remainder  ;  after  which,  you  must 
seek,  multiply  #nd  subtract,  till  you  have  brought  ihnvn 
every  figure  of  inn  dividend. 

Proof.  Multiply  the  divisor  and  quotient  togedier 
and  add  the  remainder  if  there  be  any  to  the  producl: ;  '\l 
the  work  be  right,  the  sum  will  be  equal  to  tlie  dividciid.* 

"^  Another  method  Tvbich  some  make  ima  of  to  prove  divi 
sion  is  as  follows  :  viz.  Add  th^  remainder  and  all  the  pro- 
cIUQts  of  the  56^  era?  quotient  %ares  multiplied  b.y  the  dlvisc^r 


DIVISION    OF    WHOLE    NUMBERS. 


3? 


I.  Ho^v  many  timeii is  4  tZ.  Divide  3b5G  doUuri 

contained  in  9391  ?  equally  among  8  mea. 

Divisor,Div,^uotient  Vivisor^Div. Quotient, 

4)9391(    2347  8)3656(457 

8                  4  S2 


13 
12 

19 
16 


9388 
-f  3  jRe?n. 

9391  Proof. 


45 
40 

56 
56 


SI 


3656    Proof  t^ 
additionr 


5  Eeniahider. 
Di4}isDr,Div.  Quotient. 
a9) 15359(529 
145 


Proof  by 
careers  of  9's 
5 


•x^ 


85 

58 

fi79 

261 


565)^9640(136 
S6t^ 

1514 

1095 

2190 
2190 


Bemains      18 0  llem, 

togedier,  according  to  the  order  in  which  they  stand  in  the 
work  5  and  this  sum,  when  the  work  is  right  will  be  equal  to 
the  dividend. 

A  third  method  of  proof  by  excess  of  nines  is  as  follows,  Tie. 

1 .  Cast  the  nines  out  of  the  divisor  and  place  th«  mlcom 
on  the  left  hand. 

£.  Do  the  same  with  the  quotient  aad  place  it  on  the  right 
band. 

3.  Multiply  these  two  figures  together,  and  add  their  pro- 
duct to  the  remainder,  and  reject  the  lunes  and  place  the  ex- 
cess Ht  top. 

4.  Cast  the  nines  out  of  the  dividend  and  plact;  the  esoesi 

at  bottom. 
NoTs.  S"Uic  sum  U  ilgliT,  (bo  Ttv)  fOid  boKdm  ti^wes  trfll 


so  DIVISION-  OF   "V^'UOLE   NUMBKRSi. 

DivisQr.Div.quotient.  95)85595(901 

61 )28609(469  736)86S25S(1 17^ 

47^2)25 1 1 04(532  there  remains  664 

9.  Divide  1895312  by  912.  Jlns,  2076. 

0.  U]v\(\e   '893312  by  2076.  Ans.  912» 

U.  iiivide  47254149  by  4674.       ^ns.  lOllO^Vr- 
12.  What  is  the  quotient  of  330098048   divided  6jr 
4207  ^  Jlns.  78464. 

IS.  What   is  the  quotient  of  761858465  divided  hj 
8465  ?  Jlns.  90001. 

14.  How  often  does  761858465  contain  90001  ? 

Jlns.  8465. 

15.  How  many  times  38473  can  you  have  in  119184693  ? 

^ns.  3097111^, 

16.  Divide  2B0208122081  by  912314. 

quotient  307140^^3^-^ 

MORE    EXAMPLES    FOIl   EXBRC13E, 

Divisor.  Dividend.  Remainder* 

234063)590624922((itfoiien^)8S973 
47614)327879186(  }  9182 

987654)988641654(  )  •..0 

CASE  II.  ^  ^ 
When  there  are  cyphers  at  the  right  IiailJ  of  the  tlkl- 
•or ;  cut  off  the  cyphers  in  the  divisor,  and  the  sanro 
number  of  figures  from  the  right  hand  of  the  dividend, 
tlien  divide  the  remaining  ones  as  usual,  and  to  the  re 
maindei^  (if  any)  annex  those  figures  cut  oS'from  Ijlre  dlvl* 
ffend,  and  you  will  have  the  true  remrander. 

EXAMPLES. 

1.  Divide  4673625  bv  21400. 
214{00)46736)25(2185\W^true  quatient  by  Kestitutiott. 
428.. 


S4^5  true  rem. 


COSTilAOTXONS    IN*   DIVISION.  O/ 

0.  Diade  Sr943*375  bj  6500  J^r.s.  .^8374'^' 

5.  Divide  4£1400J00  '      49000  ^^ds. 

4.  IVividc  1160911^.    !>y  89000  ..  i;2S.  ;31a  ' 

5.  Divide  9187642      b-  91700'J:;.  '^i^.  S-^    ,, 

MOHE    v. SAMPLES. 

JDlvif'   \  Di^'dend,  - 

\25000)4$(^^^'0i'^ql^otien  0 

1 20000)  I4959647t(  i/t-ir?. 

901000)6  4S47250(  )2r:?30 

720GOO)987654000(  ^.5S40^ 

CASE  ill. 

Short  Division  is  whea  the  divisor  does  Kot  exceed  12. 

RULE. 

Consider  how  many  times  the  divisor  is  contained  in 

the  firsi:  iT;;;i)re  oriiguresof  the  dividend,  put  the  result 

under,  and  carry  as  manj  tens  to  the  nesrt  ii^ire  as  there 

are  cHies  over. 

Dmde  evxry  figure  'n  the  saHie  manner  till  tlie  wliole 
is  finished. 

EXAMPLES. 

Divhm\  Dividend, 

£) 113415        3)85494        4)39407        5)94379 


^nothiit  56707 — 1 


6)120616  7)152715  8)95872  9)118724 


n)6986ig7  12)14814096  12)570196382 


Contractions  in  Division. 

When  the  divisor  is  such  a  mimber,  t3ia.tany  twofig* 
ur£S  in  the  Table,  beingaiultipiied  together  will  produce 
it,  divide  the  given  dividend  by  one  of  those  figures ;  the 
<]Uotient  thence  arising  by  the"  other ;  and  *&e  last  qua- 
Uentwill  be  the  answer. 

NoTK.  The  total  rcRiaindci' is  found  b^  ccTi.trpnng 
tiTctiist  j*(5.inriin'.!cr  by  the  fir.^  divisor,  and  t^ang  m  the 


SHrrLEMENT    TO  MULTIPLICATIOX. 


Divide  16£641  by  72. 
'i)  162641  or 


EXAMPLfiS. 


8)18071—2 

J2258— 7 


8)162641  last  rem.  7 

9)20530—1  x9 

2258—8  63 
frstrem.  -|-2 


True  ((uotient  2258f| 


by  16. 
by  24. 
by  35. 
by  m. 
by  48. 
by  54. 
by  84. 
by  108. 
10.  Divide  1575360  by  144> 


2.  Divide  178464 

3.  Divide  467412 

4.  Divide  942341 

5.  Divide    796S8 

6.  Divide  144872 

7.  Divide  937387 

8.  Divide    93975 

9.  Divide  145260 


Time  rem,  65 

Jlns.  11154 

dns.  19475-J-l 

Ans.  269244^ 

Ms.  2212j%. 

J171S.  3018;^ 

Ans.  17S59X 

Ms.  ln^i 

Ms.  1345 

Ms.  10946 


2.  To  divide  by  10,  100,  1000,  Sgc* 

RULE. 

Cut  off  as  many  figures  from  the  right  hand  oF  i\ic  divi- 
dend as  there  are  cypners  in  the  divisor,  and  these  figures 
•0  cut  of!' are  the  remainder;  and  the  otiier  figures  of  the 
•Kvidend  are  the  quotient. 


Divide  ''yGS 


EXAMPLES, 

by  10.        Ans. 


50  and  5  remains. 


:>:'*ide  5762      by  100.      Ans.  57  —62  rem. 
Oi^ide  '^65753  by  1000.  Ms.  753 —753 rem. 


vSUPFLEMSNT  TO  MULTIPLICATION. 

T?  multiply  by  a  mixt  number  rthat  is  a  whole  num- 
er  Joined  with  a  fraction,  as  Si,  S^^Qiy  occ. 

RULE. 

M*o  ilaph  by  the  whole  number,  and  take  i,  f ,  ^.  &c.  of 
fes  M^i^caiiC;  imi  add  it  tothcnrcdact. 


^¥EPL^MBNT  TO  MULTJLPLIOATION.  89 

EXAMPLES. 

Multiply  37  bySSJ.  Multiply  48  by  H 

2)37  48 

23i  ^ 

I83 
111 
74 


£4*J 

12= 

=i 

96 

1S2  Ms. 

Ms. 

106551 

•ins. 

205334 

Ms. 

6594 

Ms. 

334134 

869i  Mswer. 
h  Multiply    211  by  50i. 

4.  Multiply  2464  by     8^. 

5.  Multiply    345  by  19|. 

6.  Multiply  6497  by    5|. 

({iiesiions  to  Exercise  Multiplication  and  Division. 

1.  Wliat  will  9j  tons  of  hay  come  to,  at  14  dollars  a 
ton?  Ms.  S136i. 

2.  If  it  takes  320  rods  to  make  a  mile,  and  every  rod 
contains  5i  yards ;  how  many  yards  are  there  in  a  mile  ? 

Ms.  1760. 

3.  Sold  a  ship  for  11516  dollars,  and  I  owned  |  of  her  5 
what  was  my  part  of  the  money  ?  Ms.  S8637. 

4.  In  276  barrels  of  raisinsj  eaeh  3}  cwt.  how  many 
hundred  weight  ?  Ans.  966  cwt. 

5.  In  36  pieces  of  cloth,  each  mece  containing  24i 
yards ;  how  many  yards  infhe  whole  ?    Ms.  873  yds. 

6.  What  is  the  product  of  161  multiplied  by  itself? 

Ms.  25921. 

7.  If  a  man  spends  492  dollars  a  year,  what  is  that  per 
calendar  month  ?  Ms.  g41. 

8.  A  privateer  of  65  men  took  a  prize,  which  being 
equally  divided  amon^  them,  amounted  to  119/.  per  man ; 
what  is  the  value  of  me  prize  ?  Ans.  £77S5. 

9.  What  number  multiplied  by  9,  will  make  225? 

Ans.  25. 

10.  The  quotient  of  a  certain  number  is  457,  and  the 
divisor  8  ;  what  is  the  dividend  ?  dns.  3G56. 

1 1  What  cost  9  yds.  of  cloth,  at  35.  per  yard  ? 

Ans.  27s. 
1 2.  Wliat  cost  45  oxen,  at  Si  per  head  ?  Ans.  £360. 


40  cor.i)?oT;>'D  ADinTiON 

IS.  What  cost  s -r  •  V'h  of  tn:ligo.  at  2  dols.  SO'ctR.  nr  : 
250  cents  per  lb.  '  dns,  S360.      | 

14.  Writedown  four thou^sand-A  i  ;  v-  r.'l  an^l  ^even- 
teenj  multiply  it  by  twclvej  divide  the  product  by  nine, 
and  add  365  to  the  quotient,  then  from  that  sum  subtract 
live  thousand  five  hundred  and  twenty-one,  and  the  re^- 
mainder  will  be  just  1000.     Try  it  and  see. 


itnua^jnjiMwtn.-gt 


COMPOUND    ADDITION, 

j-S  the  adding-  of  several  nuin!)ers  together,  having  dif- 
terent  denominations,  but  of  i\v^  same  generic  kind,  io 

pounds,  shiirin<5S  and  pence,  ivr.     ''\"-:  ' —.dreds,  rpaar- 
tcrs,  &c. 


i.  Place  the  iiunibers  ^o  that  lliose  of  the  stiinc  dcnam- 
1  nation  may  stand  directly  under  eacli  other. 

2.  Add  the  first  coluiiip.  or  denomination  together,  a3 
in.  whole  nu.vibers;  then  divide  the  sum  by  as  many  of 
the  same  denomination  as  maiie  one  of  iA\Q  next  greater  t 
setting  dovrn  the  remainder  under  the  column  added,  and 
•:  rv  tlie  quotient  to  i\\e  r.ext  superior  denomination, 
.  »  v^iaii.g  tiiC  eamc  to  tiie  lust,  uiucli  add,  as  in  simplo 
.u'  i'iion. 

1.  STERLING  ]\IONEY, 

'.:,  tlie  money  of  account  in  Great-Britain,  and  is  reck- 
.:-din  Poun'fls,  Shillings,  Pence  and  Farthings,  ^o 
the  Pence  Tables. 


*  The  reason  of  this  rule  is  evidenff*:  For,  addition  of  ihls 
money,  as  1  in  the  pence  is  equal  to  4  in  flie  farthings ;  1  in 
the  shillings,  to  K^hi  the,  peacii ;  and  I  in  the  pounds,  to  SO 
in  the  shillings  ;  .cvthre  carrying  as  directed,  is  the  ar- 
ranging; the  mon;  ,  arising  from  each  coliimn,  nroperly  in 
the  scale  of  denon/iiiations;  and  this  reasoning  will  hold  goMl 
in  the  addition  of  compound  nnnd>er3  of  any  dni(yminatitp 
whatever. 


0OM?OUND   ADDtTIOX.  41 


EXAMPLES. 

£.    s     d. 

VViiat  is 

1  the  sum  total  of  47/.  135. 

"47     li>     6 

6^?.~19Z.  : 

25.   9id 

.-^14Z.  lOs.   md. 

19      2    94 

and  12^  9s.  Ud.  ? 

Thus^ 

14     10  lU 

J2      9     H 

>! 

%Slnswerf£. 

93     16    4i 

..♦f 

£.    s.    d.  qr. 

£'    ^ 

17     13 

11 

84    17    5    3 

30     1 

11     4     2 

M     10 

2 

75     13    4    2 

15    : 

10    9     1 

,    10   ir 

5 

50     17    8    2 

1 

0     12 

I       8       8 

/ 

20     10  10    1 

3 

9     8     3 

3       5 

4 

16    5    0 

4 

6    3     1 

(5) 

(6)^ 

^^>      . 

i    i^-       ^'* 

d.  qi\ 

£.    s.    d.  qr. 

£• 

s.    d,  qr 

,'   4r  17 

6    2 

7     17  10    S 

541 

0    0    0 

S      9 

10     S 

60      6    8    0 

711 

9    8     1 

.     59     17 

11     2 

7     14  11     2 

918 

16    9    3 

\  S17     16 

9     3 

18     19     9     3 

14C 

>    15  10     1 

762     19  10     1 

91     15     8     2 

300 

1     19  11     3 

407     17 

6    2 

18     17  10    3 

46 

I     10    7    3 

1     19 

9 

5      0     12 

fl 

1     14    9    3 

(8)        . 

(9) 

(10) 

£' 

s,     d. 

£.     s.    d. 

£^ 

s.    d. 

105 

17    6 

940     10    7 

97 

11     6i 

193 

10  11 

36      9  11 

20 

0     4 

901 

13     0 

11      4  10 

144 

1  10 

319 

19    7 

141     10    6 

17 

11    9 

48 

17    4 

126     14    0 

9 

le  loi 

104 

11     9 

104    19    7 

0 

15     9h 

90 

16    7 

160     10     6 

19 

9    4 

111 

9    9 

100      0    0 

234 

11  lOi 

976 

0  10 

9      0    9 

180 

14    € 

449 

12    6 

0     1*9    6 

421 

10    S| 

^0 

10    4 

12#      0    8 

841 

16    4 

M 


>^^ 


II.  Find  tlie  amoinit  of  the  foIlowing"^    f^, 
aums,  viz.  42/.  13s.  5af.— Hi.  10s. — 4/.  t 
irs.  8<^.— is;.  Os.  7(^.— .19s.   4ifZ.— sr/.f 
and  15L  6s,  J 


1 


-5?2S.  £.  IM    7    0-h 

1£.  Add  504Z.  5s.  and  Oic?.-— 34Z.  19^\  TJ.— r/.  TsM^i. 
•^247/.  Os.  llrf.— 19s.  6d.  lcp\  and  45/.  togetlier. 

Jlns.   r640  5s.  5-^i/, 

15.  Find  the  sum  total  of  14Z.  19s.  6^/.— llZ.4s.  ^d.-^ 
^51  10s. — iL  Os.  6i/.— 3/.  5s.  8«f.^l9s.  6^.  and  0.5.  Bil, 

Jus.  £60  Os.  5n, 

14.  Find  the  amount  of  the  following  snuns,  viy'. 
Forty  pounds,  nine  shillingSj     .    .    -    -     £.      .^,     a. 
Sixtj-tour  pounds  and  nine  pence,     -     - 
Ninety -live  pounds,  nineteen  shillings,    - 
Seventeen  snillinsis  and  4^fK     -    -    -    - 

o 


.fc«  /:.  201      6     l\ 


15.  How  much  is  the  sum  of 

Thirty -seven  shillings  and  six  pence,  - 

Thirty-nine  shillings^and  4M\.     -     -    -  - 

Forty -four  shillings  and  nine  pence,     -  - 

Twenty -nine  shillings  and  fliree  pence,  - 
Fifty-sinllings,   -    -    -    -    - 


^ns,  £.  10  Os.  10^/i- 

16.  Bought  a  quantity  of  goods  for  125L  10s.  paid  iW 
truckage  forty -five  sliiliings,  for  freight  seventy-nine  siril- 
lings  and  six  pence,  for  duties  tliirty-five  shillings  and  ten 
pence,  and  my  expenses  were  fifty -three  shillings  and 
nine  pence  5  what  did  t]\e  goods  stand  me  in  ? 

Ans.  jT.  136  4s.  iV, 

W.  Six  men  took  a  prize,  and  having  divided  it  equafry 
amongst  them,  each  man  shared  two  nund red  and  farty 
pounds,  thirteen  sldllings  and  seven  pence  5  how  much 
money  did  the  wliole  prize  amount  to  ? 

^  .  JUis.  £.  1444  Is.  erf. 


„^ 


lb, 

oz. 

V7Vt, 

p-. 

/A. 

oz. 

/}w5- 

frr 

16 

11 

19 

25 

8 

n 

19 

^i 

4 

4 

16 

SI 

6 

10 

IG 

8 

8 

8 

19 

14 

7 

8 

17 

21 

6 

9 

14 

17 

4 

6 

8 

ir, 

4 

fr 

10 

•^ 

9 

7 

14 

17 

0 

r 

11 

12 

•4 

9 

15 

10 

S.   AVOIJlinJPOIS    WEIGHT, 


f 

.Z5. 

lb. 

(TX. 

c^r. 

T. 

civi. 

qr. 

//;. 

QZ, 

^. 

27 

24 

15 

14 

9T 

17 

2 

24 

15 

14 

1 

17 

17 

12 

11 

19 

9 

0 

17 

10 

12 

o 

26 

28 

12 

15 

14 

15 

2 

04 

9 

n 

1 

13 

16 

8 

»r 

47 

H 

5 

19 

14 

5 

5 

15 

24 

10 

12 

(^ 

00 

t 

00 

00 

12 

-2 

16 

U 

12 

12 

T7 

19 

5 

27 

15 

H 

4.    APOTHECARIES    \S' EIGHT. 


5  9  sr. 

.a 

?   3  ^»*- 

fe 

S 

f>  9  ^r. 

9     1     17 

7     2     19 

V^ 

11 

G     I     15 

S    2      9 

6 

5     0     i:'. 

4 

9 

7     0     12 

6     1     17 

i 

ii    1       7 

9 

10 

1     9,     16 

4     0     16 

9 

3     2     12 

4 

8 

1     2     13 

5     2     12 

6 

1     0     If) 

9 

0 

0     1     10 

e  1    10 

0 

6     2     19 

4 

9 

2     1       G 

5. 

CLOTH    JfEAS 

URD. 

I          .                               Hi 

ji.   qy,7i(L. 

J».  JB.  5ri\ 

m. 

JEJ.F.  qr.tja. 

n    3    0 

44     5 

o 

84   e   1 

Is     2    1 

45    4 

5 

07     1     5 

10    0     1 

06    2 

o 

76    0     2 

42    3     5 

S4     4 

1 

,52     2     3 

57    2    2 

07    0 

0 

55     2     2 

49    2    2 

61     2 

I 

09     2    5 

6.    DRY   MEASURE. 


^Icqt.pL 

?;?*.  pk\  qt. 

iiU.lpk.qLp, 

1     T    1 

17    ^    5 

9,5    ^    7    X 

£     6    0 

Q4    9.    7 

64    2    6    1 

15    0 

13    3     6 

43    0    4    0 

2     4     1 

16    5    4 

52    3    5    1 

2     G    1 

sr    2    6 

94-    2    3    0 

S        6       0         T.r 

56    0    7 

54    3    r    0 

f 


yds.fL  in. 

//.'• 

4     2     11 

f; 

5     1       S 

1 

1     2      9 

o 

G    2     10 

1 

1     0       G 

8 

3     17 

0 

f  .    WINE     MP^ASURE. 


/r^i. 

,0f.  2-'"-^' 

hh^. 

■C"?.  f/f.  ^f. 

iun»hhd,gaLqt. 

:>9 

3     I     S 

3     1 

34      2    34    2 

1  r 

2     1     2 

2    0 

19      1    59    1 

£4 

3    0     1 

9 

14     0     i 

28     2      2    1 

1^^ 

i     ''     '  * 

0 

0     C     1 

19     a    3-2    2 

8 

0 

16 

3r    3   n   1 

40 

': 

"t 

0      1      0    G 

8.    LONG     MKASUREi 


m.  jOir. 

■??o. 

46     4 

16 

58     5 

23 

9     6 

34 

ir   4 

18 

r   3 

15 

5     2 

24 

U. 

m.fur. 

pc. 

86 

2    6 

32 

52 

1     7 

16 

64 

2     5 

19 

*  -.7 

1     4 

15 

'•r 

2    3 

25 

28 

2     4 

17 

9,    LAT;D   or    SqUARE    MEAStrilE. 


acregtrco: 

Is. rods. 

acTPS^-i 

roods»rods. 

^rv./>. 

5?,W. 

478      3 

s: 

odS 

2 

IS 

5 

136 

816      2 

J  / 

U) 

3 

00 

6 

129 

49       1 

27 

9 

1 

39 

8 

134 

(?5       S 

34 

1 

3 

00 

0 

146 

9      3 

S7 

0 

2 

27 

4 

54 

10.  Si>LII>    MPASIiRli:. 


41 

{t 

3 

^ 

15 

1446 

19 

43 

4. 

114 

le 

172Qf 

49 

6 

7 

as 

3 

868 

4 

2r 

10 

127 

14 

£34 

. 

n. 

TIME, 

r. 

«p. 

w. 

(ffl. 

Vr. 

^ 

A, 

m.   &at. 

5r 

11 

S 

6 

24 

536 

23 

54  34 

5 

0 

e 

3 

2J 

40 

12 

40  CA 

^ 

8 

e 

5^ 

13 

119 

14 

09  17 

40 

10 

2 

4 

14 

9 

11 

18  14 

10 

7 

1 

S 

8 

24 

8 

16  15 

IS.   CIUOULAn  MOTIOI/. 

S.^fti                  S.     ^  '  "^ 

5  29  17  14  11  29  59  59 
I  6  10  17  0  00  4D  10 
4  18  17  11        0   4  10  49 

6  14  18  10        4  11  6  10 


COMPOUND  SUBTRACTIOK, 

X  EACHES  to  find  flie  diftcrejicc>  ^j^equalityor  crces^j 
betwetiTi  any  two  sums  of  diverse  dtfaominatiojiB.. 

RULE. 

Place  those  numbci^  under  eacli  other,  \i  hich  ars  cl 
the  same  denomination,  the  less  l>eing  below  the  greater ; 
^egin  with  the  least  denomination,  and  if  it  exceed  fiie 
figure  over  it,  borrow  as  many  units  as  make  one  of  the 
liext  ereater ;  subtract  if  tlicrefrom ;  and  to  fbc  diftArcnrx 
add  tne  upper  fi;5ure,  remembering  always  to  add  one  to  tko" 
Ocxtsypcrinr  defimnlmition  f(ir  Ihjrt  wni^rhyfTU  b(Trc<nvc>J. 


Note.    Tne  method  of  proof  is  the  samea*  insimple 
fiubtraction. 


From 
'I'ake 

KXAMPLE* 

1.  Steiibi";  Momu, 

£,  s,  d.qr,            £.  s.  (Lqr. 
346  16  5  3               14  14  6  2 
123  17  4  2               10  19  6  3 

94  11  6 
36  14  8 

r:eH>. 

£•7  19  1  1 
ved     44    10  2 

36  11  a 

RS 

lid 

r.     5.     d. 

'^5     0    0 

4  19  11 

Borrov 
Paid 

Lent 
Received 

Due  to  m( 

7  1112 
4  17  3  1 

£,  s,  d.  gr 
36    0  8  2 
18  10  7  3 

Kemal 
unpj 

% 

Tate 

£.   s,   d,  qr. 
476  10  9  1 
277  17  7  1 

I^ein. 

From 

Take 

141  14     9  2 
19  13  10  2 

(10) 

£.  s.  d. 

125  01  8 

124  19  8 

(11) 

C  s.  d.qr 

10  15  7  1 

0     9  6  3 

Kcm. 

12.  i5onowed  27Z.  Ms.  and  paid  19?^.   175.  ^d.  how 
;iu.c?5  remains  due  ?  Jim.  £7  ISs.  6rf. 

VZ.  llow  jniich  does  3I7Z.  6.^-.  exceed  Vi^l.  18^.  5^^.  ? 

^/2-s.  £138  75.  6^a. 
11,  Frcm  eleven  pounds  take  eleven  pence. 

Ms.  £10  19s.  \d. 
15.  From  seven  tliousand  two  liundred  pounds,  take 
>y^  Us.  GyL  Ms.  £7n\  9.S.  5id. 


COMPOUND   SUiri'KACYXOK.  •^T 

16.  How  much  docs  seven  hundred  and  eight  pounds, 
exceed  tldrty-nine  pounds,  fifteen  shillings  and  ten  pence 
halfpenny?  ^ns.  ^668  4s.  Ud. 

17.  From  one  hundred  pounds,  take  four  pence  half- 
penny. Ans.  £99  I9s.  7 id, 

18.  Received  of  four  men,  the  following  sums  of  money, 
viz.  The  first  paid  me  57^.  lli\  4rf.  the  second  25Z.  16s. 
7d.  the  third  19Z.  14s.  GJ.  and  tlie  fourth  as  mucli  as  all 
the  other  three,  lacking  i9s.  6d,  I  demand  the  whole 
sum  i^gipived  ?  Ans.  y^  165  5s.  4d, 


Frmn 
Take 

lb, 
6 

oz. 

11 

3 

O 

pXvL 
14 
16 

IROY    WEIGHT. 

oz.pwtgr.        lb.  ox.  imit.gr. 
4     19     21          44     9       6     19 
2     14     23          17     5     16     IS 

Rem. 

Ih. 

684 
683 

ox 
2 

i 

pwt. 

10 

9 

c:;*      ih,    ox.  plot,   p\ 

14  ^      942     2      0         0 

15  892     9       2 

• 

Ih,  oz. 
7      9 
3     12 

dt\ 

12 
9 

A  V  vO  I K  D  U  r  0 1 S  E    W  B I G  H  T 

5     i     15             n     12     i      1^0     9 

P    ' 

T.    cii'i*.  qr,  lb.     oz.  di\  T.    ctvi,  qr.  lh.%z.  dr. 

810     11     0  .20     10     11  Sir     12     1    12  19    12 

193     17     1     20     12     14  180     12     1    14  |o    14 


AV0TKEC?IK1ES^   WEIGJlT. 


19       8     7  4     1       37  iS5     7    5      I      14 

9     11     6  1     £      15  17    10    6     1      IB 


48 


CQ.MiV)u':^l>    SVBTKAfl^J.O>J(? 


5*.   OLO'fH   MEASURE. 


13.  qr.n^ 
55  1  £ 
l|  1  3 

E.E.  m\ 

467  S 
291  5 

1 

C 

76S     1     5 
X49  £  t 

y2,  (7r».4i2i?!, 

513  5  1 

ir4  i  0 

E.E.  qi'. 
615  0 

£26  2 

7ia. 

1 

0 

E.FL  qr,m. 

&l.  ^1\  q^ 
14     3     4 


6.   DRY   MT.ASUKiiN^- 

8     15  -^   17    fi    5    0 

3   16  e  2  e  i 


0l  ^.  ^.  ^. 
£10    0     1 


:4   2   1 


r.   VINE    M:^.iSXJHE. 

/iftt?,  gal.  qi.  fit.      T.hhd.gcd.  qtph 
13      0     3     0  £    3     20    3     1 

10   6051         ie£roo 


hhd.  gtll.  qt.  pt. 

G12    £3     1     0 

7:5     37     1     1 


4    £     11     0 

e  £  11   1 


2,7    1  '6    67 
19    £    4     SD 


JJuL 
5£1 
256 

14 
£5 

5#.  ,;/. 
£  1 
3  0 

8.   LONG    MEASURE. 

in.  fuT.pc. 
41     6    £2 


10 

6 

£5 

le, 
16 
10 

1 

fur. 
3 

3 

5 

.J 

U:,  m.  far. pa. 
86  2  6  S£ 
24     1     7     31 


15. 

VI. 

fdr. 

ITO. 

9 

£ 

0 

r 

1 

1 

1 

J 

9.  LAND   OR    SqUARB    AQCAStJlOi:. 


JK    roods,  rods. 
^9        1        10 

CA        1        25 


Jl.    r.  po. 

s^fi.sg.iu. 

29  e  17 

599   181 

17  1  56 

19   1S3 

•tf,      ^r.    r^d^.  ^.   qr.  rods,  sg.fi.  sq.ixu 

--^      0      25  ISO     1     10  gfiO        84 


rg 


119      1      37  49  ^1     11  143      la 


ID.   SOLID  MCASXJRJET^ 


tans,  jfif . 
116  24 
109    59 


canf  5.  /if. 

tm^  S^ 

vt. 

72  114 

45     18 

140 

61  120 

16  14 

145 

11.  Tj5ifr> 
4B     11    S    5  14    356    20    49    19 


yr9.  mo,  w.  (tit,  yr$.  dcnjs,  li.  wi^  seSi 

f^     11     S     1  24    352    20    41     2p 


WUJ2. 

w. 

d. 

h. 

set. 

472 

2 

15 

18 

4Q 

218 

4 

16 

29 

54 

w.    d.   h.  rtdru  szc* 
781     1      8  j^    21 

197    S*t|2fp    53 


l2.  OlBCLaAR  JIOIIOJ^. 

•;sr.   •     '     •                iSr.   •  '     • 

9    23    45    54                    9    29  34    54 

3       7    40    55                    r    29  40    36 

■•        "I          •"                   llif-«                                                                   ■■!       ■     i^»M    II  I  ■   I       II 


/y 


QUESTIONS, 

S'howing  the  use  of  Compound  Mdiiimi  and  Suhtraclii^ 

NEvV-YORK,  MARCH  22,   IS  14. 

3.  Bought  of  George  Grocer, 

12  C.  Sqrs.  of  Sugar,  at  52s.  per  cwt.        £  S3  13    0 

2  8  lbs.  of  Rice,  at  3c?.  per  lb.                         '^0  7    0 

5  loaves  of  Sugar,  Art.  35lb.  at  Is.  Id.  per  lb.  1  1/11 

S  C.  2  qrs.  14lb.  of  E^iisins,  at  36s.  per  CAvt,    o  I^    6 

£41      5    S 


2.  Wiiat  sum  added  to  17 Z.  lis.  S^d.  will  make  lOOL? 

^^KS.  8£Z.  8s.  3(^.  3r/?-* 

S.  Borrowed  501.  10s.  paid  again  at  one  time  17^."  i !«. 
6i.  and  at  another  time,  9^.  4s.  Sd.  at  another  time  7L  - 
9s.  6^.  and  at  another  time  19s.  did.  how  much  remains 
unpaid  ?  Ms.  £  15  4s.  9i(/. 

4.  Borrowed  lOOZ.  and  paid  in  part  as  ibllov/s,  viz.  at 
otte  time  9.11.  lis.  6d.  at  another  time  19Z.  17s.  4kL  at 
another  time  lOdpUars  at  6s.  each,  vrad  at  another  time 
two  English  guineas  at  28s.  each  and  two  pi  star eens^  at 
I4irf.  eacii ;  how  much  remains  due,  or  uiipaid  ? 

Ans.  £52  12s.  S^J.    < 

5.  A,  B,  and  C,  drev/  tlicir  prize  money  as  foilov.s,  viz. 
A  had  f5l.  15s.  4.d.  B  iiad  "^threc  times  as  much  as  A, 
lacking  15s.  6d.  and  C,  had  just  as  much  as  A  and  B  bcih} 
pr^y  how  ranch  had  C  ?  Jlns.  £S02  5s.  lOd, 

6.  I  leHt  Peter  Trusty  1000  dols.  and  afterwards  lent 
him  26  dols.  45  cts.  more.  He  has  paid  me  at  one  tiaic 
361  dols.  40  cts.  and  at  another  time  416  dels.  09  cts.  be- 
sides a  note  which  he  gave  me  upon  James  Paywcll,  for 
145  dols. -90  cts. :  hov/ stands  the  balance  between  us  ? 

Jns.  The  bcdance  is  ^105  06  cis.  due  to  me, 
r.  Paid  A  B  in  full  for  E  F's  bill  on  me,  for  10 JL  lOff. 
v;z.  Ij^ave  him  Richard  Drawer's  note  for  loL,  14.?.  9i» 
l\'ter  tfohn son's  do.  for  SOL  Os/^Cid.  an  order  on  llobert 
j)oaler  for  S9L  lis.  the  rest  I  mahC  up  in  cash.  I  wai\| 
c'j  know  ^^liat  sum  will  make  up  fhe  denclencj  ? 

dns.  £^  s:..  ^iCt 


'^- 


ULl'lPLlOATlON 


r.  \  nicrcuaut  iind  six  debtors,  M'ho  togct:ier,  owed  him 
9,91; l.  105.  6rf.  A,B,  C,  D,  and E,  owed  him  1675/.  ISs 
Qd.  (S  it;  what  was  F's  debt?    Ans.  £1241  16s-.  9iL 

9.  A  merchant  bou|[^ht  17('.  5-qrs.  14lb.  of  sugar,  of 
vhidi  he  sells  9C.  oqrri.  25lb.  how  much  of  it  remains 
unsold  ?  y*- "  .^ws.  7C.  ^qrs.  \7lh. 

\{),  From  a  fasliionable  piece  of  cloth  vvhich  contained 
52j(is.  2nn.,  a  taylor  was  ordered  to  take  three  suits,  eacli 
6j('s.  5:(|vs.  how  mucli  remains  of  the  piece  ? 

Jins,  S2yds,  2qrs.  2na. 

1   .  The  war  between  England  and  America  commen- 
ced a^-i^ril  19,  ,1775,  and  a  freneral  peace  took  place  Jan 
uarj  -wOlh,  17S;1:  Low  'oiiji;  did  the^  war  continue?         • 

JiiiSs  Tyrs,  9 mo.  Id, 

COMJPOUND  MULTIPLICATION. 

Co:!iIPOUND  Multiplication  is  when  the  Multiplicand 
consists  of  several  denominations,  &c. 

T.  To  Midti/ply  Federal  Money. 

RULE. 

iSIultiplj  as  in  whole  numbers,  and  place  the  separa- 
trix  as  many  figures  from  the  right  hand  in  the  product, 
as  it  is  in  the  multiplicand,  or  given  sum. 

-^         '  EXAMPLES. 

•^  S  cts.  g  f7. cm. 

1.  Multiply  S5  09  bv25.    2   Multiply  49  0  0  5  by  9r, 

25   "  .97 


1754b  54S0S5 

7018   ^  ^  441045 


Frod.   S87?,  4^  •  S4753,  4  8  5 

^..Multi'}^^'  1     doi.  4  cts.  by  505    Ms.  317,  20 

4.  iMuitiply  41  cts.  5  mills  by  150    Ms.    62,  25 

5.  Multiply  9  dollars  by  50  Ms.  450,  00 
€.  ?.lultiply  9  cdlmhy  50  Jtns.  4,  5a 
r.  Multiply  9    miflTby  50    Ms.      0,,  45 


3.  There  were  forty-ane  men  concerned  in  tke  paj 
incnt  of  a  sum  of  money,  and  each  paid  3  dollars  ana  9 
mills ;  how  much  was  paid  in  all  ? 

Jlns.  gl23  SGcfs.  9mills 

9.  The  number  of  inhabitants  in  the  United  States  i5 
ilvf)  millions ;  now  suppose  each  should  pay  the  trifling 
sum  of  5  cents  a  year,  for  the  term  of  12  years,  towards  a 
continental  tax ;  how  many  dollars  would  be  raised 
tftcreby  ?  "^ 

Atts,  three  millions  Dollar f, 

£»  'H  JTultijph  tlie  Denominations  cf  Sterling  Monevj 
Weights f  Measures^  Sjz* 

RULE.* 

Write  down  the  Multiplicand,  and  place  the  quantity 
Gnderneath  the  least  denomination,  for  the  Multiplieip^ 
and  in  multiplying  by  it,  observe  the  same  rules  for  carry- 
ins  from  one  denomination  to  another,  as'nn  Compouxdl  ' 
Audition. 

INTBODUCTORY    EXAMPLES 


/:. 

s.  d,  q. 

s.d. 

Ml!} 

tjply  1 

11  6  2 

hj3. 

How  much  is 

S  times  11  8 

5 

/;. 

s. 

a. 

s 

Vr 

£^ 

17  8  £ 

1  15  5 

d. 

, 

s.    d. 

'\5 

10 

8 

2 

9A 

12 

G 
5 

n 

15    S 
4 

15     n     10  10     16    4  SI     10    9J 

5  0  7 


®  When  accounts  are  kept  in  pounds,  shillings  and  penCCj^ 
(his  kind  of  mdtiplication  is  a  concis^mid  elegant  method  of 
irnding  tlie  value  of  goods,  at  so  muflper  yard,  ib.  &c.  the 
g^Tn^S  v\i^  biytrf;  to  mitltipi^  fbe  glveffprico  by  the  qxiantity 


flOMrOUA'D    MULTlI'UiBATifN.. 

6. 

SI 

IG 

8 

1:3    17     10 

14     10 

ri 

S 

9 

10 

32 

12 

10 

G     19     1 

26    8 

4J 

u 

12 

12 

Fr  act  leal  Questions. 

W!:at  cost  nine  yards  of  cloth  at  5s.  6d.  per  yard  ? 
£0  5  6  price  of  one  yard.  ' 
Tvluliiply  by    '  9  yards. 

Jiiis,  £  2  9  6  pric(?  of  nine  yards. 

r.     «^.  d.  £.    s.  d. 

4  Gallons  oi  v/inc,        at  0    8    7  per  gallon.  1     14  4 

5  C.  Maiaj;a  Raisins,  at  1    2    t^  per  c^vt.    5     11  5 

7  re:i:n.i  of  paper,         at  0  17     9^  per  ream.  6      4  GJ 

8  yds.  of  broadcloth,    at  1     7     9^  per  yard,  li      2  4 

9  lb.  of  cinnamon,        at  0  11     4i^perlb.        5      2  2| 

11  tons  of  hay,  at  2     110   per  ton.    23      0  2 

12  bwshels  ofapples,     at  0     1     9  per  bush.    1       10 
12  bushels  of  v/heat,      at  0    9  10  per  bush.    5     18  0 

2.  Wlicn  the  multiplier,  that  is,  the  quantity,  is  a  com- 
posite number,  and  greater  than  12,  take  any  two  such 
numbers  as  when  muitiplicd  together,  will  exactly  pro- 
duce the  given  quantity,  and  multiply  first  by  one  of  those 
figures,  and  that  product  by  the  other  5  and  the  last  pro^ 
duct  ^viU  be  tlie  answer. 

EXAMPLES. 

■\Vliat  cost  28  yards  of  cloth,  at  6s.  lOd.  per  ysrd  ? 

0    6     10  price  of  one  yard. 

'^'  '  ''^ly by  7 


2    7    10  price  of  7  yards. 


*^ 


Mver,  £  9    11    4  jj^ice  of  28  jaidSr 

5* 


cq:^: i^Ou m :>  Min,rjivi;,]cxTiOK . 


<•* 

il. 

gr5. 

^ 

yards 

at 

7 

4 

3  per  yard,  = 

27 

— 

at 

9 

10 

0 

— ,               ~: 

44 

— 

at 

12 

4 

2 

-— 

5a 

— . 

r>A 

8 

S 

1 

s= 

7^ 

— 

^f 

19 

A  1 

0 

— «•                     =2 

£D 

^_ 

^^ 

3 

6 

55 

-—                     = 

&4 

._ 

at 

LS 

4 

2 

<—                     = 

fi3 

. — , 

at 

11 

9 

0 

*—                =:= 

G3 

-^  at 

£-i 

17 

6 

0 

— —                ra 

H4 

---  at 

1 

4 

o 

0 

•—               rr: 

AKswjs;:;^. 

£• 

.^.     tf. 

8 

17    6 

33 

5    6 

£7 

4    6 

22 

14  lOj 

71 

14     0 

3 

10  10 

77 

3     6 

56 

8    0 

118 

2    G 

174 

0     0 

3.  ""."/hen  no  two  numbers  muUiplied  together  will  ex- 
actly make  the  nvaltipriGr,  you  must  multiply  by  any  two 
^rliose  protluct  will  come  the  nearest ;  then  multiply  ijhe 
Upper  line  by  what  rcm?Jned ;  vrliich  added  to  the  last 
product  gives  the  answer. 

EXAMPLES. 

^'^Imt  will  'u  yds.  of  clotli  come  to  at  \7^?M,  per  yd.  ? . 

0     \7    9  price  of  1  yai'd, 

Multiply  !>y  5 

Produces     4      3    9  price  of  5  yards, 
J^IuUId'v  bv  9 


PrwliJccs  39     18    9  price  of  45  yards* 
1     15     6  price  of  2  yard^* 

*5;:t5s:-c:%  /: 41     14    S  price  of  47  yards. 

CttrES'no:rs.  an'^WSRS* 

25  ells  of  linen,     at  0    3  0^  per  ell. 

!7  ells  of  dov/liicj  at  0    1  6^  per  ell. 

\  d9  cwt.  of  siigar,    at  3  10  6    per  cwt 

*  52  yds.  of  cloth,     at  0    5  9    per  yd. 

'  X9  ibs.  of  indi«o,    at  0  11  6    per  lb. 

^  £9  yds.  of  caviibric.  at  0  IS  7    per  yd. 

^tll  vd^.  broad dojh,  at  1     S  6  « per  yd. 

gS^  U^ve  h>'?L'^j       set  1    9  4    a|^ccB> 


4 

1 

5\ 

1 

6 

^i 

137 

9 

6 

14 

19 

0 

10 

18 

6 

19 

IG 

11 

124 

17 

6 

m 

t? 

a 

4^  To  find  ific  value  oi'  a  inniCH  c^i  wc^^i*,!",  hy  Viaviifv 

[  fhc  price  of  one  I'O'mci. 

I  If  the  pice  bo  farthings,  muitiplv  2*.  4t].  by  the  far- 
{hin*;?  in  the  j>rlce  of  one  lb. — Or,  if  the  j.'nccl>e  nence, 
mnltiplj  9.S.  4d.  Ijy  tiic  pence  in  the  price  of  one  li).  and 
in  either  case  tlie  'product  will  be  the  cms'.ver. 


What  will  1  cv;t.  of  rice  cometOja.!  2^d.  per  Ih.  ? 

112  fi!rthings«=2    4  price  1  cwt.  at  ^<1.  per  lb. 

9  faithings  in  tlic  price  of  1  ib, 

^nfi.£l  1     0  price  of  1  cwt.  at  9  v  per  Ib. 
What  will  1  cwt.  o(  lead  coine  to  at  Td.  ner  iV? 


^n^,  £3     5     4 

(liiestions,  Jlnswers, 

1  cwt.    at  2^  per  lb.  =  £1    S    4 

1  ditto,  at  2|d  -«     =     i     5     8 

1  ditto,  at  Sd  —=180 

1  ditto,  at-2d  --.     =     0  18    8 

1  ditto,  at  5  id  -»    c=     1  12    8 


Examf.es  of  Weights,  Measures^  i^'t^ 
1  KcTVTnuchfs  5  times  Tcwt.  3q?^.  15  Ib.  r" 
Cwt.  ijrs.  lb. 
r      5     Id 
5 


4ff»s.  Cwt 

.  S9 

.1 

19 

lb  oz. 

7?u*#. 

i^r. 

( 

:7^'f. 

qr. 

?&. 

oz. 

S.A1 

uttSj)!j  2D  2 

7 

isb 

4 

7  4. 

P) 

o-^ 

I 

13 

12 

p.r.; 

»r!  Pa  80  9  10   . 

c4 

I^. 

1G4 

0 

£6 

a 

/r*' 

COMrOTJKD    MULTlVMCA  (  ION, 


pr™', 

ANSWERS* 

yds.  (jr.  na. 

7/is.  qi\  na^ 

4.  Miiiti])l/i4     3     2  bv  11 

163     2    2 

hhd.  -.  qt.  pt. 

hhd.  g.  qt.  ft. 

5.  ]\fiiltlply  21  i J     C     1  by  32 

254  61  2     0 

le.  r.i.jiw.  fo. 

le.  m.fiir.  jt>*. 

6.  Multiply  81  2    G    21  by  S 

6j5  1     4      8 

^i.    r.  p. 

«.    r.  |7. 

7.  Multiply  41    2   U  by  13 
yr,  vu  IV.  fi. 

748     0    38 

?/r.  721.  ic\  d. 

8.  Multiply  20    5    3     6  by  14 

23G    5     2     0 

S.    "^      '     '/   " 

fS',     ^       '     " 

5.  Multiply   1     15  48  24  hv  5 

r    19   2    0 

cds.fL 

cds.  ft. 
29     5^ 

10.  Multiply  S    87  by  8 

Fraciical  ^^iicstions  in 
WEIGHTS  &  MEASURES. 

1.  What  is  the  weight  of  rhlids.  of  sugar,  each  weigh- 
ing 9  cwt.  3  (ps.  12  lb.  ?  Ms.  69civL 

2.  What  is  the  weight  of  6  chests  of  tea,  each  weigh- 
ing 3  cwt.  2  qrs.  9  lb,  ^  Ans.  Qlcwt.  Iqr.  26Z6. 

3.  How  much  brandy  in  9  casks,  each  containing  41 
gals.  3  qts.  1  pt.  ?  Arts,  i^76gals.  Qqts.  Ipt. 

4.  In  35  pieces  of  cloth,  each  measuring  27^  yards, 
how  many  yards  ?  Ans.  971yds,  Iqr, 

5.  In  9  fields,  each  containing  14  acres,  1  rood,  and 
25  poles,  how  many  acres  .^    Jlns,  129a.  Qqrs.  QSrcds. 

6.  In  6  parcels  of  wood,  each  containieg  5  cords  and 
96  feetf  how  many  cords  ?  *^ns.  S4icords, 

7.  A  gentleman  is  possessed  of  H  dozen  of  silver 
spoons,  each  weighing  2oz.  15  pwt.  11  grs.  2  dozen  of 
tea-spoons,  each  weighing  10  pwt.  14  grs.  and  2  siivtar 
tankards,  each  21  oz.  15  pwt.  Pray  what  is  the  weight 
of  i\\Q  whole  ?  M9,  Bib,  IQq».  ^wt  ^^. 


GOMPOmrND  DIVISION; 

X  EACHES  to  find  how  often  one  number  is  conUincd 
in  another  of  different  denominations. 

DIVISION   OF    FEDERAL    MONEY. 

ptr*Any  sum  in  Federal  Money  may  be  divided  aS  ^ 
whole  number ;  for,  if  dollars  and  cents  be  written  down 
a»  a  simple  number,  the  wliole  will  be  cents  ;  and  if  tin; 
sum  consists  of  dollars  only,  annex  two  cyphers  to  tlic 
dollars,  and  the  whole  will  be  cents ;  hejice  the  follo^/^ 

GENERAL  IIUI.E. 

Writedown  the  given  sum  in  cents,  and  divide  as  In 
vh^l^umbers ;  tao  quotient  will  be  the  answer  in  cents* 

Note.  If  the  cents  in  the  given  sum  are  less  than  1(^ 
you  must  always  place  a  cypher  on  their  left,  or  in  tll^ 
(en's  place  of  the  cents,  before  you  write  them  down. 

EXAMPLES. 

>    1.    Divide  55  dollars  68  cents,  by  41. 

41)3568(87    the  quotient  in  cents  |  and  when  fiiere 

S28  13   any  considerable   remainder,  you 

~—  ^nav  annex  a  cypher  to  it,  if  you  pleasfij 

288    ,      and  divide  it  again,  and  you  v/itl  hiTve 

2£r  ^he  mills,  &c. 

R»!m.       I 

S.  Divide  Gt  dollars,  5  cents,  by  14. 

14)2105(150  cents=l  dol.  50  cts.  but  to  bring  cents 
14  into  dollars,  you  need  only  poiiit  off  two 

figures  to  the  right  hand  for  cents,  arlfl 

70  the  rest  will  be  dollars,  &c.     . 

5 
n.  Divide  4  dols.  9  cts.  or  409  cts.  by  G.    ^ns.  68  ctT,^, 
4  DiYide.  9  d^s;  9A  cts..  by  1 P^  4^S;.  7Z  cts^' 


$.  Divide  97  dols.  43  cts.  by  85.      J?/is.  gl  14cfs.  6?fi,  . 

6.  Divide  248  dols.  54  cts.  by  125.  >^ 

^ns.  198cis.  8m.=:Sl  98cfs.  8m.  ; 

7.  Divide  24  dols.  65  cts.  by  248.  Am,  9cts.  9w. 

8.  Divide  10  dols.  or  1000  cts.  by  25.  Ans.  4Qcts. 

9.  Divide  125  dols.  by  500.  Ms.  25cts. 
10.  Divide  1  dollar  into  33  equal  parts.        Ms.  Gcts,+ 

PRACTICAI.  QUESTIONS. 

1.  Boiiglit  25lb.  cf  cCiTce  for  5  dollars  5  whatis  tliat  a 
pound?  Ans    ^Ocfs. 

2.  if  151  yards  of  Iribh  linen  co3t  49  dob.  '  '  "^t 
is  that  per  yard  ?                                           A, 

3.  If  an  cwt.  (f  sugar  cost  8  dols.  96  cts.  %v 

per  pound  ?  Jlnt  ^f0t$. 

4.  If  140  reams  of  paper  cost  329  dols.  ^vliat  is  tnat 
per  ream  ?  Ms,  S2  Socts, 

5.  If  a  reckoning  of  25  dols.  41  cts.  be  paid  equally 
amon^  14  persons,  what  do  they  pay  a  piece  ? 

Ms.  ^\  Sleets, 

6.  If  a  man's  wages  are  235  dols.  80  cts.  a  year,  v/hat 
is  that  a  calendar  month  r  Ms.  Sl9  65cis.     * 

7.  The  salary  of  tlie  President  of  tlie  United  States,  is 
tvr<intv-five  thousand  dollars  a  vear :  what  is  that  a  day  ? 

Ms.  S68  49cfs. 

2.  Tf>  divide  the  denominations  of  Sterling  Moneij^ 
Weights^  Measures y  S[c. 

RULE. 

Begin  Vv'itli  the  highest  denomination  as  in  simple  di- 
vision 5  and  if  any  thing  remains,  find  hmv  many  of  the 
next  lower  denomination  this  remainder  is  equal  to; 
which  add  to  the  next  denomination ;  then  divide  again;, 
carrying  the  remainder,  if  any,  as  before;  and  so  on^  till 
the  whole  is  finished. 

Frohf — The  s:fmc  as  ia  Simple  Division. 


COMPOUND   DIVISXON.v 


59 


Divide 


97 


£XAMFL£5. 

12    2  by  5. 


Quo't.  £19     8      9    2 


3. 

4. 

5, 

6. 

7. 

8. 

9, 
10. 
11. 
12. 
13. 
14. 
15, 


£• 

Divide  31 

Divide  22 

Divide  70 

Divide  56 

fitvide  61 

Divide  24 

Divide  185 

Divide  182 

Divide  16 

Divide  1 

Divide  6 

Divide  1 

Divide  943 


11  6  by 

3  9  by 

10  4  by 

11  5i  by 

14  8  by 

15  6^  by 
17  6  by    8 

16  8  by    9 

1  11  by  10 
19  8  by  11 

6  6  by  12 

2  6  by    9 
11  6  by  12 


£•    ^'    ^* 

8)27    18    6 

£3      9    9} 

JiViS.  15  15  9 

7    7  11 

17  12  7 

11     6  Si 

10    5  9} 

3  10  9i 

23    4  81 

20    6  Si 

1  12  2J 

0    3  7i 

0  10  6^ 

0     2  6 

79    0  llj 


2.  When  tlie  divisor  exceeds  12,  and  is  the  productof  two 
or  more  niiinbcrs  in  the  table  multiplied  together. 

Divide  by  one  of  those  numbers  first,  and  the  quotient 
by  the  other,  and  the  last  quotient  will  be  the  answer. 


4. 


Divide  29 
Divitle  27 
Divide  67 
Divule  24 
Divide  128 
G.  Divide  269 
7  Divide  248 
R.  Divide  65 
■\  Divide      5 


EXAMPLES. 

^5  0  by  21 
16  0  by  52 

9  4  by  44 
16  6  br  36 

9  0  l)y  42 
12  4  by  56 
10  8  by  64 
14  0  by  72 
10  3  by  81 


£• 

.  s,d. 

Ms.  1 

8  4 

0 

17  45 

1 

10  S 

0 

IS  Oi 

5 

]  2 

4  16  3i 

3 

17  8 

0 

18  3 

0 

14J 

d. 

£*• 

sr. 

({. 

0  by    SO. 

1 

6 

g 

6  by  lOS. 

1 

5 

4 

6  by  121. 

1 

13 

6 

0  by  144. 

0 

4 

9 

60  rcp»(pot3S^o  Dtv;sigj^^ 

£.    s. 
<I0.  Bivide  115  10 

11.  Divide  136  16 

12.  Divide  202  IS 

13.  Divide    34    4 

3.  When  tlie  divisor  is  large,  and  not  a  composlto 
tfUiaber,  you  may  divide  by  the  whole  di\isor  at  (Jnce> 
afj;^  raanner  of  long  division,  as  follows,  viz. 

EXAMPLES* 

Divide  128Z.  Us.  ^d.  by  47. 
r.    s.    d  £.    5.    d. 
4!r)lS8  13    3(2    ]4    9   quotient 
94 

54    pounds  remaining. 
fftUUiply  by    20  and  add  in  the  13s« 

t        producreB    693   Siiilllngs,  ^vliicu  divided  by  4/",  givqsi 
^  47  [14s.  ill  the  quotient. 

223 
183 

35  shillings  remainiugt 
Multiply  by     12  and  add  in  the  Sd. 

ptoduees    4S3  pence,  ivliich  divided  as  above^  gives 
'23  [9d.  in  the  quotient. 


.1^^ 


£.  Divide  115 
5.  Divide  85 
4.  Divide  315 
IT.  Divide  132 
e;  Divitle  740 
7.  jDivHe  S^o 


5.      .'/. 

£• 

5. 

<;; 

13    4    by    31. 

Ms,  3 

13 

4 

6    S    by    75. 

1 

0 

9 

3  10^  by  355. 

0 

ir 

SJ 

0    8    Uv    G8. 

1 

IS 

9J 

16    8    l>y  IDO. 

>?• 

s 

<J 

IS  to    hv    95: 

ti 

■'^ 

v.* 

EXAMPLES    OF 

WEIGHTS,  MEASURES,  &c. 
1%  Divide  14  cwt.  1  qr.  8  lb.  t)f  sugar  equally  ainQBg 
8  m^. 

0.    gr.  lb.  oz. 
8)14     18    0 


1    S    4    8  Quotient* 
8 


14    1     8    0  Proof. 

5a  Divide  6  T.  11  cwt.  5  qi's.  191b.  by  4 

JIns.  IT.  I2cwt  Sqr$.  2,5lh.  l^ox. 
G.  Divide  14  avt.  1  qr.  1£  lb.  by  5 

Ms.  SLcwt.  S^.  13/6. 9(7r.  9ir-*-f. 
4.  Divide  16  lb.  IS  oz.  10  dr.  by  6 

*5ns.  2Z&.  12or.  15Jr. 
ff.  DivJdfe  56  lb.  6  oz.  17  pwt.  of  silver  into  9  equd 

6.  Divide  £5  lb.  1  oz.  5  pv.i:.  by  24 

*/?ks.  1Z&.  loz.  ^pwt.  l^y, 
r.  Divide  9  hhds.  £8  gals.  2  qts.  by  12 

Jji.'?.  Ohhds,  49gals.  2qls.  Ipt^ 

8.  Divide  165  bu.  1  pk.  6  qts.  by  So 

Jins.  4ku  Spin's.  2iiU. 

9.  Divide  17  lea.  1  in.  4  fur.  21  po.  by  Gi 

10.  Divide  43.  yds.  1  qr.  1  na.  by  li 

JIns.  2vds.  Cqrs,  Snn, 

11.  Divide  9rE.E.  4  qrs.  1  nii.  by  o 

12.  Divide  44  galldns  of  brandy  aqua! ly  aiaoni^  144 
ffDldiers.  •  dns.  i£ill  a-plecc. 

15.  I#ougiit  a  <lozen  of  silver  spoons,  wluch  togctl.er 
weighed  3lb.  £  oz.  13  ])v:t  12 '^rs.how  much  sliver  did 
eath  sgoou  contain  ?  Jlns,  Soz.  Anivt.  1 1,p'. 

14.  Bou^jht  17  cwt.  8  qrs.  19  B).  of  sit;;:ir,  and  sold  cut 
oireth?i-dof  it}  how  much  iiJinains  nnsold  ? 

6 


<»2  •oMi'OUMD  liivisiby, 

15.  From  a  piece  of  cloth  containing  64  yards  2  ita, 
a  taylor  was  ordered  to  make  9  soldiers'  coafts,  which 
took  one -third  of  tlie  whole  piece ;  how  many  yards  didi 
each  coat  contain  ?  Jlns.  2,7jds,  Iqr,  2na. 


PRACTICAL  QUESTIONS. 

I.  If  0  yards  of  cloth  cost  4L  Ss,  7 id.  what  is  that  per 
yard  ? 

9)4    ^    7    2 


9    3    2  Answer 


9.,  If  U  tons  erf  hay  costSSZ.  Cs.  2(f.  what  is  th&tper 
ton?  *^ns.  £2  Is.  lOi. 

3.  IF  12  gallon:^  of  brandy  cost  AU  15s.  6c?,  wliat  is  thtf 

per  gallon  .^  Jins,  7s,  lid,  ^qrs. 

4.  If  84  llis.  of  cheese  cost  i^.  l6s.  9cf.  what  is  that  per 
p<fund  ?  Ans.  5 la, 

5.  Bought  48  pairs  of  stockings  for  11/,  2s,  how  mu(ih 
1  pair  do  tlicy  stimd  me  in  ?  Jtns,  4s.  7hd, 

6.  U  a  recKoning  id'  5L  8s\  lOjid,  be  paid  equally  among 
13  persons,  ',vliat  do  they  i'i:ij  a-picce  ^      Jins.  8y.  4^^.^. 

7.  A  niece  of  clolli  containing  24  yards,  cost  IS/,  6/. 
U'l^at  did  it  cost  per  yard  ?  Snis,  15.«.  3^. 

8.  If  a  hogshead  ol"  wijie  cos^t  53^.  12s.  what  is  it  a 
gallon  ?  ^  Jfns,  10s.  Si, 

9.  If  i  ewt.  of  sugar  cost  3Z.  10.?.  wiiat  is  it  per  pound? 

Jilts.  7ld. 

10.  If  a  man  spends  7ll.  14s.  6d,  a  year,  v/liat  is  that 
ptr calendar  month?  .^^ns. /J5  19s.  G^d, 

11.  The  IVincc  of  ^\'alcs'  saJaiy  is  150,000/.  ayearf 
wiiat  is  that  a  day  ?  Jns.  £ 4 1 0  1 9s.  24. 

12.  A  privateer  tako3  a  prize  worth  12,465  dollars,  of 
which  ihQ  owner  takci  enc-half,  xhQ  oficcrs  one-fourth 
and  ilie  remainder  is  equally  divided  among  ihi  sailors 
who  are  125  in  number  •  hew  much  is  each  sai lor-a  part  ? 

^nir  g24  9Sa/§. 


ir/,Dv 


13.  ']'liro<>,  marciKims,  A,  II,  and  ('.  lave  a  sliip  ^.li 
company.  A  hath  -J,  B  |,  and  C  J.  and  lliey  receive  far 
ft-eight  '22o!,  IGs.  oJ.  it  is  required  to  divide  it  among 
tlieowncr^  accord in£^  to  tl;eir  respective  shares. 

Jhi^.JVs  Hhare  £N3  Os.  5cL  IPs  share  £57  4s.  2a. 
C's  share  £28  12s.  Id. 

14.  A  jirivaieer  haviu';  taken  a  pnzc  v.-nrth  S6S50,  it 
is  divided  into  one  hundred  shares;  of  which  the  cap- 
tain is  to  have  11:  2  lieutenants,  each  5;  12  midship- 
men, each  2 1  and  the  remainder  is  to  be  divided  equally 
among  tiic  snilors,  wlin  twd  105  in.  number. 

J:is,  CapUiurs' share  S75::  fAkts,  Lkid^^.  Sn342  :A^cfs. 
amidsJiijrmans  S^37,  and  a  sailor's  ^ooS^cts. 


REDUCTION, 

X  EACUES  to  brino-  or  change  numbers  from  one  name 
to  another,  without  altering  their  value. 

Reduction  is  either  Descending  or  Ascending. 

Descending  is  when  great  names  are  brought  into 
small,  as  pounds  into  sliillings,  days  into  hours,  &c.-- 
This  is  done  by  Multiplication. 

Ascending  is  when  small  names  are  brought  into  creat, 
as  shillings  into  pounds,  hours  into  davs,  &c.  Tills  ii 
performed  by  Division. 

REDUCTION  DESCENDING, 

RULE. 

Multiply  the  liigliest  <]enomJ nation  given,  by  so  many 
cf  the  next  less  as  make  one  of  tliRt  greater,  and  thus 
continue  till  you  !iave  brought  it  down  as  low  as  your 
question  requires. 

Proof,  Change  the  order  of  the  question,  and  divide 
your  last  prciiuct  by  tl;e  last  multiplier,  and  so  on. 

EXAMPLES. 

t ,  I\\  &3L  ioL  9d.  Sqm,  haw  many  faithings  ? 


u4  REDUCTION. 

£r    s.     d.  qrSs 

S5    15    9    £  Proof. 

20  4)24753  Ms.  24758 

515  sliillings*  12)6189  2qrs. 

12  

2|0)51j5  9d. 

6189  pence.  

4  £25  15  93 

£4758  fari;hiug&. 

y^TY..  Jn  mi?)Jp]yh:-  by  50, 1  aildcd  in  the  I5s,^t>y 
X^  tlic  P<L— auii  I;v  -  •■;  >  ^lors,  wliich  iiuist  always  b^ 
done  in  like  cases. 

^.  In  31^.  lis.  lOi.  Iqr.  how  many  farthings  ? 

.i/i^^  30329 

5.  In  46/.  5s.  lief.  S^rs.  how  many  farthings  ? 

Ans.  44447 

4.  In  til,  125.  how  many  shillings,  pence  and  far* 
(hings  ?  *5ks.  1232s.  U7S4d.  59lS6qrs. 

6.  In  84/.  how  many  shillings  and  pence  ? 

^ws.  1680s.  20160 J, 
C  In  18s.  9d.  how  many  pence  and  farthings  ? 

Ans.  9.9.5a.  ^QOqrs. 

7.  In  512Z.  8s.  5d,  hov/  many  half-pence  ? 

Ans.  149962 

5.  In  846  dollars  at  ^s,  each,  how  many  farmings  ? 

1^.^25.  243^48 
i9.  In  41  guineas  at  28s.  cacli,  how  many  ncnce.^ 

Ans.  TSr76 

10.  In  59  pistoles,  at  22s.  how  many  shillings,  penc^^ 
dnd  fartliings  ^ 

Jlns.  1298.9,  IjjTGd.  62S04gr5. 

11.  In  87  half-johaniies,  cil  -?.<^.  h(n7  many  shilling 
SiX-penccs,  and  three-pence?^  P 

Ans.  1776s.  $^552  six-pemca:  7'iOA  three-pences. 

12.  In  121  French  ciT.wriS,  at  Gs.  ScL  each,  how  many 
(fence  and  fartliings  ?  .-^  .'Z?:-^.  [)mod.  Sgrr^Ogrs- 


HEDlUCTION  ASC£NDJ«G, 
HULE. 

Divide  tlic  lowest  denomination  given,  by  so  many  of 
that  name  as  make  one  of  the  next  higher,  and  so  on 
tlirough  all  llie  dentminations,  as  far  as  your  question 
npmiires. 

rRoov.    Multiply  inversely  by  the  several  divisors. 

KXAMPLES. 

1.  In  ^24765  fai*things,  how  many  pence,  shillings  ani 
pounds  ? 
Fa rtl lings  in  a  penny  =  4)224rC5 

Per^e  in  a  shilling       =  12)56191  1 

Shillings  in  a  pound     =  2!0)468|2  Td, 

'/:234   2s.  rd.  Iqr. 
Ans.  5619U7.  4682s.  2S4t. 
NoTi:.  Tlie  remainder  is  always  of  the  same  name  as 
^tlie  dividend.  « 

r.  Bring  50329  farthings  into  pounds  ? 

Jlris.  £Sl  lis.  lOd,  Iqr. 
S.  In  4444r  fartiiiugs,  how  many  pounds  ? 

.6ff?s.  £46  5s.  lid.  Sqrs 

4.  In  59136  farthings,  how  many  pence,  shillings,  and 
pounds  ?  Ans.  UTS4d.  1232s.  £61  12s. 

5.  In  20160  pence^liow  m^any  shillings  and  pounds  ? 

.^ns.  i680s.  or  £84. 
G.  In  900  fartliings,  how  many  pounds  ? 

Ms.  £0  ISs.  9d. 
7.  Bring  74981  half-pence  into  pounds  r 

Ms.  £156  4s.  9,hL 
3.  lu  243648  farthings,  how  many  dollars  at  6s.  Ciicii.^ 

Sms.  B846. 
9.  Reduce  15776  pence  to  guineaSj^at  283.  per  guinea* 

Jlns.  41   --. , 
ID.  In'  6^^04  ftr^.hings,  how  many  pistoles,  at  22s. 


U.  Ja  ri04  tiu:Q^e-pejices,ho\7manjhaIirjohatit[e$^5^t 

12.  la  SSrso  ftrtliings,  how  many  French  crowns,  at 
69.  8d.  ?  ^Ks.  121. 


Reduction  Ascending  and  Descendhig 
1.   MONEY. 

1.  Tn  ISl?.  Os.  9^d.  how  many  half-pence  ? 

e^ns.  58099 

2.  In  58099  lialf-pencG^  how  many  pounds  ? 

Am.  iSlL  05.  9^^. 
S.  Bring  SSrGO  half'pence  into  pounds.  *^ws.  £49  105, 

4.  In   £14L  Is.  Sd.  how  many  shillings,  six-pences^ 
tTireepences,  and  farthings?  "      Ans,  4f281.s.  856a|sf or- 

pences,  in 25  ihree-pences^  and  ^05500  farthings. 

5.  In  isn.  how  many  pence,  and  English,  ^r  French 
browns  at  6s.  8d.  each  ?         Ans,  S2880rf.  41 1  crowns, 

G.  In  249  English  half-crowns,  how  many  pence  and 
gounus  ?  Jns.  996{)d,and£4:l  10s. 

7.  In  545  guinCi^j  at  21  s.  each,  how  many  shilling?, 
goats  and  pence  ?    Ans,  7266^?.  21798  grHs  and  87192ff* 

8.  In  48  guineas,  at  28s.  each,  how  mai5v  4^d.  pieces  ? 

"  Ans,  3584 

9.  In  81  guineas,  at  278.  Ad.  each  how  many  pounds  ? 

Ans.  £110  145. 

10.  In  24596  pence,  how  many  shillings,  pounds  and 
Jjistoles  ?  Ans,  20S5s.  £10\  135.  and  92  pistoles. 

1)5.  over, 

11.  In  252  moldores,  at  T^Gs.  each,  how  many  guineas 
at28s.  each.^  '  Ans,  524. 

12.  in  J  680  Dutch  guilders,  at  £s.4d.  each,  how  many 
pistoles  at  22s.  each  ?  Ans.  178  j>?*,se;o6V5,  4s.    ^ 

15.  Borrowed  1248  English  crowns,  at  6s.  8d.  eachjj 
how  many  pistarcens,  at  14 ^d.  each  will  pay  tlie  debt  ? 
Ans.  6885  pistarcens  a:id  7^. 
14.  In  50?.  how  many  shillings,  ninc-pcuces,  six-peri« 
cpSjfQur-pfinccs,  and  pence,  and  of  each  ancqua  number  r 
\9.d.Ar^d.^M*'r^d.-\'ld.  =C2fZ.  and  £50  r^ 
XSJ000(^rfy52==^'S7.T  A)\}i 


ili:,DUCTiON  OF.  FEDERAL  MOXI^Y. 
t.  Reduce  Cr45  dollars  into  cents. 


2745  dolhi-s  "^ 
100 


^ins.  274500 


Here  I  multiply  by  100,  the 
cents  in  a  dollar ;  but  dollars  are 
>readily  brought  into  cents  bj  an- 
nexing two  cyphers,  and  into 
mills  by  annexing  three  cyphers. 
Also,  any  sum  in  Federal  money  may  be  wi-itten  down 
as  a  wliole  number  and  expressed  in  its  lowest  denomina- 
tion ;  for,  when  dolliirsand  cents  are  joined  together  as 
a  whole  number,  without  a  scparatrix,  they  will  sliew 
Iiow  many  ecu  Is  tlic  frivcn  sum  contains;  and  when  dol- 
lars, cents,  i^nd  mills  are  so  joined  together,  i]\(^y  will 
p'vew  tlie  whole  number  of  niills  in  the  given  sum. — 
Ilj'uce,  properly  speakiiig,  tiiere  is  no  reduction  of  this 
moii'^y :  for  ccntr>  are  readily  turned  into  dollars  by  cut- 
ting olV  tl^iC  two  riglit  hand  figures,  and  nVills  by  pointing 
aft' tliree  fjgiircs  with  adot :  tlie  figures  to  the  left  hand 
of  the  dot,  aiiMlolIars;  and  tlie  figures  cut  ofl*  are  cents, 
orcenU  and  mills. 
2.  In  345  dollars,  how  many  cents  and  mills  ? 

J71S.  S4500cts.  345000  mills, 
5.  Reduce  43  dols.  78  cts.  into  cents.        ^Ins,  4878 

4.  Reduce  25  do!s.   8  cts.  into  cents.        Jinsi,  2503 

5.  Re^luce  54  dols.  56  cts.  5m.  into  mills,  w^ns,  54365 

6.  Reduce    9  dols.  9  cts.  9m.  into  mills.    Ans,  9090 

.  S    cf^. 

r.  "R^iduce  419£!5  cents  into  dollars.      Ans,  419  £5 

^.  Change    4896  cents  into  dollars.  48  96 

9.  Change  45009  cents  into  dollars.  450  09 

V>.  Bring       4625  mills  into  dollars.  4  63  5 


2.   TKOY   V/EIGHT. 

K  How  many  grams  m  asHvcr  tankard j  t^f     *.«^nf 
Ub.  no.-,  15  pwt? 

4  d       ■  - 


•  /  lb.    9%,  j>wt 

^  1     11     15 


REDVQTfOlC. 


X     11     15 

IS  ounces  in  a  pound. 

23  ounces. 


20  pennyweights  in  one  oun^c. 

h  .  — 

475  pennyweights. 
£4  grains  in  one  penwjweigJit. 

1900 
950 

Eroof.    24)11400  grains.    .Ins. 
2,0)47.5 
(12)23  15  pwt. 

1  lb.  11  oz,  15  pwt, 

^.  In  246  Qz.  Iiow  many  pwts.  and  grains  ? 

Ms.  49^0pivt  llSOaO^TVv 
5.  Bring  46080  grs.  into  pounds.  Jlns.  8 

4 .  In  97397  grains  of  gold  how  many  pounds  ? 

Ms.  16lh.  IOdz.  ISpivt.  5grs. 

5.  In  15  ingets  of  g©ld,  each  weigliing  9  oz.  5  pwt. 
how  many  grains  ?  JIns.  66600 

6.  In  4  lb.  1  oz.  1  pwt.  of  silver,  liow  many  table 
spoons,  weighing  23  pwt  each,  and  tea-spoi?ns,  4  pwt. 
6  grs.  each,  can  be  made,  and  an  equal  number  of  each 
sort  ? 

^Spwt.+^pict.  6^rs.=:654gTS.  the  diviser  and  4lh. 
loz.  lpwt.^25544grs.  the  dividend.  Therefore  25544 
^654=36  Jlnsiver. 


S,    AVOIRDUrOlS   WEIGHT. 

in  89  Qwt.  S  ^rg.  14  lb.  12  qz.  hsw  man^r  ounces  ? 
4 

^9  q*art6rs.  £S|ftriIed  Uf .] 


S59cp&>tei'3*  Proof. 

S8^-  16)161068 

^876  28)10066  \Qox* 

^19  

4)359  14Z&. 

10066  poumls.  — 
16                  BOcwt  Sgrs.  14C&.  12qx. 


60398 
10067 


161068  ounces.    Answer. 

B.  Ih  19  lb.  14  oz.  11  dr.  how  many  drams  ? 

Ms.  5099. 
5.  In  1  ton  how  many  drams  ?  Ans.  573440. 

4.  In  24  tons,  17  cwt.  3  qrs.  17  lb.  5  oz.  how  many 
«Ilices?  Ans.  892245. 

5.  Bring  5099  drams  into  pounds. 

Ans.  19lb.  Uoz.  lUr. 
6i  Bring  573440  drams  into  tons.  Ans.  1. 

7.  Bring  892245  ounces  into  tons, 

Ans.  24  tonsy  ITcwt.  Sqrs.  I7lb.  5oz. 

8.  In  12  hhds.  of  sugar,  each  11  cwt.  25lb.  how  many 
pounds?  Ans.  15084. 

9.  In  42  pigs  of  lead,  each  weigiiing  4cwt.  3qrs.  how 
(Oany  fother,  at  19cvvt.  2qrs.  ?    Ans.  W  father ^  A^cwt. 

lOyA  gentleman  has20hhds.  of  tobacc.^  each  8cwt. 
3qrs.  14lb.  and  mshes  to  put  it  into  bo:  35  contai-iing 
TjQtb.  each,  I  demand  the  number  t    -,0  <*«      muat  get  ? 

Ans.  384. 


4.     AVOTHECAtiiEs'   Av;:i.j;    : 

t^  la  9f5  B§  15  23  lOgrs.  how  many  grains. 

Ans.  5579^. 
£.Mn5- 55799  graitfS,  how  many  pounds  ^ 

•^rs.  9ft  85  15  29  19^r.      . 


5,    CI.OTII    MKASUnE> 

1.  In  95  yards,  howinanv<»uiirters  and  nails? 

2.  In  S41  yards,  Sqrs.  Ina.  how  many  nails  ? 

Jna.  54G9. 
S.  In  3783  nails,  how  many  yards  r 

JJjzs.  23Giffh.    l<7r.  3 ??/■/. 

4.  In  61  Ktis  English,  how  many  quarters  and  nails  ? 

5.  In  56  Ells  Flenilsli,  hov,'  niany  quarters  tLJid  nails? 

6.  In  148  Eiis  English,  how  many  Ells  Flemish? 

Anfi.9.4CjE,F.  2qrs. 

7.  In  i9£0  nails,  \io\y  many  yards,  Elis  Flemish,  an(J 
EllsEneilish? 

Ans,  im;ds.  16QE.  R  andSSE.E. 

8.  How  many  coats  can  be  made  out  oP  36|  yards  dF 
I'-oadcloth,  allowing  If  yards  to  a  coat  r  Jlns.  21 


6.    DS.Y   MEASUriE. 

1.  In  136  bushels,  how  many  pecks,  quarts  and  pints  ? 

Ans.  544pks.  4S52qis.  STMpts. 
S.  In  49  biisli.  3pks.  5qts.  how  many  quarts  ? 

Ms.  1597. 

3.  In  8704  pints,  how  many  bushels  ?        dns,  136. 

4.  In  1597  quarts,  how  many  bushels  ? 

driii,  49bus,  Spks,  5qts. 

5.  A  man  wouhl  sldp  720  bushels  of  corn  in  barrels, 
which  Vvriilhold  S  bushels,  3  pecks  each,  how  many  bar- 
rels must  he  get  ?  Ms,  192. 

r.    V;iNE    MEASURE. 

1.  In  9  tuiis  cf  wine,  how  many  hogslicads,  gallom 
and  quarts  ? 

Ans,  ^6hhds,  22(jSgal  D072gf5. 

2.  In24hhds.  18  gals.  2qts.how  many  pints  ? 

Jh:s.  12244. 
5.  In  9072  quarts,  hew  many  tuns  ?   '  Ans,  9* 

4.  In  1905  pints  of  wine,  how  many  hogsheads  ? 

Ans.  Shhds,  49sah.  fr/f. 


J:  In  17S9  quarts  of  cider,  how  many  barrels  ? 

Ans.  Ubls.  QoqtS. 

6.  What  number  of  bottles,  containing  a  pint  and  a 
hftlf  each,  can  be  filled  with  abarrcl  of  cider  ?  dns.  168. 

7.  How  many  pints,  quarts,  and  two  quarts,  each  an 
caual  numbers  may  be  filled  from  a  pipe  of  wine  ? 

*  Ms.  144. 

8.   LONG   MEASURE. 

1.  Ill  51  miles,  how  many  furlongs  and  poles  ? 

Ms.  40Sfur.  l6S20poles. 

2.  In  49  yards,  how  manv  feet,  inches,  and  barley 
corns  ?  Ms.  i47ft.  \764inch.  59,92b.c. 

3.  How  matiy  inches  from  Boston  to  New-York,  it 
b^iqg.£48  mile?  ?  Ms.  157lS2SQinch. 

4.  Ift  4352  inches,  how  many  yards  r 

Ms,  mOyds.  9ft .  Sm 

5.  Xn  682  yards,  how  many  rods  ? 

Ans,  682x2-j-ll=124ro/f5. 

0.  Ih  15840  yards,  how  many  miles  and  leagues  ? 

Ms.  9ra,  Slea. 
?.  How  many  limes  will  a  carriage  wheel,  16  faet  and 
Scinches  in   circumference,  turn  round  in  going  from 
New-York  to  riuladelphia  ;  it  being  96  miles  ? 

Ms.  30261  tiines^  and  Si  feet  over. 

8.  How  many  barley-corns  will  re«ich  round  the  glob^ 
itbeing560  degrees  r^  Jns.  4755801600. 

9.    LAND    Oil    SqUv^-lUi    MEASURE. 

1.  In  241  acres,  3  roods  and  25  poles,  bow  many  square 
rods  or  perches  ?  Ms.  SSTOSperdies, 

2.  In  20692  square  polps,  Iww  maiiy  acres  ? 

Jhis.  129(/.  if.  !%•. 
5.  Ifa  piece  of  land  contain  2  i  iicres,  anii  an  inclosuft 
Cf  17  acres,  3  roods,  and  20  ro^^s  be  taken  out  of  it,  how 
mSDy  perches  are  there  in  the  i  e.aainder  ? 

Ms.  980  jierche-p. 
4.  Three  nekls  coiitsiii,  the  iri/^t  7  acres,  the  second  10 
acres,  the  third  12  acres,  1  rood  ;  how  many  shares  caa 
tlievbe  dividcu  inti?,  eacli  sliare  Lo  contain  76  rods  ? 

MX  61  shaye^ciiid4i  jWs^Jrer^ 


^S 


10.   SOLID    MEASURE. 

1.  In  14  tons  of  hewn  timber,  how  muny  solid  ittchea? 

Ms.  14x50x1728=1209600. 
.*i.  In  19  tons  of  round  timber,  how  many  inches  ? 

Ms.  1313280. 
~.  !ii  21  cord3  of  wood,  how  many  solid  {e^t  ? 
Ms.  21x128=2688. 

4.  In  12  cords  of  wood,  how  many  solid  fe^t  and  inches  ? 

^ns.  15S6ft.  and  2654208fnc7^ 

5.  in  4608  solid  feQt  of  wood,  how  many  cords  ? 

11.    TlAfK. 

!.  In  41  weeks,  how  many  days,  hours,  minute?,  ahii 
seconds  ? 

Jiis.  QSrd.  6888/;.  413280mi?.  aiid  £4796^,^^:. 

2.  In  2l4d,  15h.  31m..  25s^.  how  many  9et>ondii  ^ 

A)iS,  185454SjS«i% 

3.  In  24796800  seconds,  how  many  v/eeks  ? 

Ms.  41  Tfeelcx 

4.  In  184009  minutes,   how  many  days  ? 

Ms.  lQ7d.  ISli.  49min. 

5.  How  many  days  from  the  birlhof  Cliilsf,  to  Chvist- 
uras,  1797,  allowing  i\iQ  year  to  contain  365  (Lap,  6  hours. 

Ms.  656S5^id.  6h. 

6.  Suppose  your  age  to  be  16  years  and  £0  (lays,  how 
ruiny  seconds  old  are  you,  allowing  365  days  and  6  hours 
•?>  tlie  yaar  ?  •3tis.  5u6649600sec.  ^ 

7.  !«Vom  March  viJ,,  to  November  19th  followiiig,  in- 
clusive; how  many  days?  Ms.  262. 

12.    ClUCULAR   MOTION. 

1.  In  7  signs,  15^  24'  40"  how  many  di*^recs,  miniiles* 
and  seconds  ?  Ms.  225*»  1S524'  cutd  811480" 

2.  .Bring  1020300  seconds  into  signs. 

Ms.  Q-slgns^  13^  25' 

qUKSTIONS  TO  EXERCISK    REDUCTION'. 

1.  In  1259  groats,  how  many  farthings,  pence,  fclilirO":! 
am!  gifjntas  a;:  2Ss.  .5725;r£0X4i5rs,  d030i< 

•  ^  "^.  ^ J.  ffKfiJ  U^uirieass  27s.  S^ 


2.  i^orrowed  10  English  guineas  at  £Ss.  each,  and  24 
English  crowns  at  6s.  and  8d.  each ;  how.many  pistoles 
at  22s,  each  will  pay  tlie  debt  ?  ^ns.  20.  ^ 

3.  Four  men  brought  each  17L  10s.  sterling  value  in 
gold  into  the  mint,  how  many  guineas  at  21s.  each  must 
they  receive  in  return  ?  Ans,  66  giiin.  14s. 

4.  A  silversmith  received  three  ingots  of  silver,  each 
weighing  27  ounces,  with  directions  to  make  ti  em  into 
spoons  of  2  oz.  cups  of  5  oz.  salts  of  1  oz.  and  snuiT  boxes 
01  2  oz.  and  deliver  an  equal  number  of  each  5  what  was 
the  number  ?  Jim.  8  of  eachy  and  1  oz,  over, 

5.  Admit  a  ship's  cargo  from  Bordeaux  to  be  250 
pipes,  130  hhds.  and  150  quarter  casks  [i  lihcis.]  how 
many  gallons  in  all ;  allowing  every  pint  to  be  a  pound, 
what  burden  was  the  ship  of?  »§ks.  44415  ^c^s.  tmd 

the  skip^s  burden  vms  158  tons,  12cirf.  9>qTS, 

6.  In  15  pieces  of  cloth,  each  piece  20  yds.  how  many 
French  Ells  .^  Jins.  200, 

7.  In  10  bales  of  cloth,  c:ich  bale  12  pieces,  and  each 
piece  25  Flemish  Ells,  how  many  yards  -^    •Ans,  2250.    ^ 

8.  The  forward  wheels  of  a  waggon  are  14^  feet  11^ 
circumference,  and  the  hind  wheels  15  feet  9  inclies,  how 
many  more  times  will  the  forward  wheels  turn  round  than 
the  hind  wlieels,  in  running  from  Boston  to  New-York, 
it  being  24S  miles  ?  Ms,  71^7. 

9.  How  many  times  will  a  ship  97  feet  6  inches  long, 
sail  her  length  in  the  distance  01  12800  leagues  and  ten 
yards  ?  Ans.  2079508. 

10.  The  sun  is  95,000,000  of  miles  from  the  earj'x, 
ami  a  cannon  bail  at  its  first  discharge  fiieS  about  a  nuie 
in  7^  seconds  ;  how  long  would  a  cannon  ball  be,  ,'».t  i:i*it 
raie  in  flying  from  h^re  to  the  siui  ? 

Am,  2%r.  2l6t?.  Vlh,  ^"Jhn. 

1 1 .  Tlie  Sun  travels  through  H  signs  of  the  Zodiac  in 
half  a  year  ;  how  many  degrees,  minutes  and  seconds  ? 

dns,  liiQdeg.  i0S00mi?i.  648000s«c. 

12.  Ilow  many  strokes  tlo^s  a.  •  ^^gular  clock  strike  in 
565  days,  or  a  year  ?  ms,  56940. 

13.  ilow  lon^j  will  it  take  to  '^ruul  a  miliioii  at  fiierate 
of  5^;  amintite  ^      .to.5.33^.     ;;^t    cr  XSd  'ZMi.'lOnu 


14.  Tk«  national  debt  of  England  amounts  to  about  SI9 

Bijlli'^V"'?   >f  pounds  r^terling ;  now  long  would  it  take  to 
>l,t  in  dollars  (4s»  6d.  sterling)  reckoning 
i-iv^sioii  twelve  hours  a  day  at  the  rate  of  50 
r\tty  und  So5  days  Iq  tlie  jear  ? 
Ans,  94  yearsM  1S4  dai^s^  5  hours,  20  min. 


FRACTIONS. 

Jf  RACT1ONS5  or  broken  nambers,  are  expressions  fo? 
any  assignable  part  el  an  unit  or  whole  number,  and  (in 
general)  are  oi  two  kinds,  \iz. 

VULGAR  AND  DECIMAL. 

A  Vidgar  Fraction^  is  represented  by  two  numbers  pla- 
ced one  above  another,  with  a  line  drawn  between  thenij 
thut,  J,  |,&c.  Signifies  three-fourths,  five-eights,  &c. 

The  figure  abcrve  the  line,  is  called  the  numerator,  and 
jlSthat  below  it,  the  denominator, 
Thus       $  5^ Numerator. 
'      ^  8  Denominator. 

The  denominator  (which  is  the  divisor  in  division) 
show,s  hovv  iTiSLh  Y  parts  the  integer  is  divided  into ;  and  the 
nur?.ei*ator  (whkh  j>^  !  erernaiiider  after  <livision)  shows 
how  many  of  t jose  |>ai^s  are  i»ieantby  therfraction. 

A  traction  u  said  to  be  in  its  least  or  lowest  terms, 
when  it  is  expressed  by  the  least  numbers  possible,  as  | 
when  reduced  to  its  lowest  terms  will  be  i,  and  ^  i% 
equal  tv»  l-,  ^c. 

PROBLEM  I. 

T©  abbreviate  or  reduce  fractions  to  their  lowest  teana 
RULE. 

Divide  th^  terms  of  the  given  fraction  by  any  numbei 
which  will  divide  thein  without  a  remainder,  and  the  quo- 
tients again  in  the  same  manner ;  and  so  on,  till  it  apnear^ 
^lat  there  is  no  number  greater  than  1,  which  mil  diTiJe 

^^5  and  tht  fraction  will  be  in  its  least  t^ns\ 


JPEAOTIOKS.  ^^ 

EXAMPLES. 

1*  R^uce  444  to  its  lowest  terms. 
(5)  (2) 
6)|44=7#=^=T  the  Answer. 
2.  Reduce  f||  to  its  lowest  terms.       dnstvers  h 

5.  Reduge  f  ^|  to  its  lowest  terms.  i 

4.  Reduce  ^Yt  *^  ^^^  lowest  terms.  -^ 

5.  Abbreviate  f  f  as  much  as  possible.  |4 

6.  Reduce  |f f  to  its  lowest  terms.  f| 

7.  Reduce  |4t  *^  ^^^  lowest  terms.  § 

8.  Reduce  ^^^  to  its  lowest  terms.  | 

9.  Reduce  \^l  to  its  lowest  terms.  ^ 
10.  Reduce  li^  to  its  lowest  terms.  | 

PROBLEM  II. 
To  find  the  value  of  a  fraction  in  the  known  parts  of 
#he  integer,  as  to  coin,  weight,  measure,  &c. 
RULE. 
Multiply  the  numerator  by  the  c-ommon  parts  of  the 
jXteger,  and  divide  by  tlie  denominator,  &c. 

EXAMPLES. 

What  is  the  value  of  f  of  a  pound  sterling  ? 

Numer.    2  • 

20  shillings  in  a  pound. 

5enom,  5)40(155.  4rf.  Ans. 


10 
9 

1 
12 

—  • 

5)12(4 
12 
2.  What  is  the  value  of  -J|  of  a  pound  sterling  ? 

^lins.  Ifes.  5d»  2^yr5. 
5.  Reduce  |  of  a  shilling  to  its  proper  quantity. 

•-ins.  4id. 
4..  What  is  the  value  of  |  of  a  shilling  ?  *>iiis.  Md 
5,  AVhat  is  the  value  cf  H  of  a  powid  troy  ?  An?.  9oa. 


6   How  inucli  is  .^  oi  an  hundred  '.veiglit  .^ 

Jlns.  Sqrs.Tlb,  lO-f^QSS* 
r.  What  is  the  value  of  f  of  a  mile  ? 

Ms.  Gfur.  QGpo,  1  \ft. 

8.  How  much  is  J  of  an  cwfc.  ? 

.Urn,  Sqrs,  Sib,  loz,  l.'2^^r. 

9.  Reduce  |  of  an  Ell  Englisii  to  its  proper  quantity. 

Ans.  '^.qrs.  S-lnn, 

10.  How  much  is  f  of  a  hlui.  of  wine  ?   Jns,  54gaL 

11.  What  is  the  value  of  ^-  of  a  day  ? 

Jus.  ]Gh.  SGmin.  55 ,^-^ sec, 
^  PROBLEM  III.        ^       ^       ' 
To  reduce  any  given  quantity  to  the  fraction  of  any 
greater  denomination  of  the  same  kind. 
RULE. 
Reduce  the  given  quantity  to  the  lowest  term  mention^ 
ed  for  a  numerator:  then  reduce  the  integral  part  to  the 
samef  term,  for  a  deoominatori  which  will  be  the  frac- 
tion required. 

EXAMPLES. 

1.  Reduce  13s.  6d,  2qrs.  to  the  fraction  of  a  poutjd. 

20  Integral  part 15  6  2  given  sum, 

12  19. 

240  162 

4  4 

960  Denominator-       650  Num.     Jns.  f f §~if/;. 
£.  Wliat  nartof  an  liMudred  weidit  is  Sqrs.  14lb.  .^ 
Sqrs.  \4lb.=mb.        dns.  -^,%=:^i 

3.  What  part  of  a  yard  is  Sqrs.  Sna.  ^  Jlns,  \  5- 

4.  What  part  of  a  pound  sterling  is  13s.  4d.?    An^,  -|. 
dH  What  part  (; ^^  3  weeks.  4  days  r 

6.  What*part  oi  a  mile  is  tlur.  ^2r 
fur.  2>o*  yd,  ft.    fed^ 

6     £6     3     2r=:4400  Num. 

a  mile  =5280  Denora.  Ms.  f||S-=|' 

r.  Reduce  7oz.  4pwt.  to  tne  fraction  of  a  pound  tror. 

8»  What  part  of  an  acre  is  2  roods,  20  poles  ?  Jln^*  g 


%  Reduce  54  gallons  to  tlic  fraction  of  a  hogshead  of 

wine.  •      •^'^^-  ^ 

*.  What  part  ©f  a  hogshead  is  9  gallons  ?     Jins.  \ 
Ih  What  part  of  a  pound  troyislOoz.  lOpwt.  l^'S;/ 

BECIM  AlT  fractions. 

A  Decimal  Fraction  is  that  whose  denominator  is  an 
unit,  with  a  cypher,  or  cyphers  annexed  to  it,  Thus,  ^, 

The  integer  is  always  divided  either  into  10, 100, 1000, 
&c.  equal  parts  ;  consequently  the  denominator  of  the 
fraction  will  always,  be  either  10,  100, 1000,  or  10000,  &c. 
nvhich  being  understood,  need  not  be  expressed  5  for  the 
true  value  of  the  fraction  may  be  expressed  by  writing 
the  numerator  only  with  a  point  before  it  on  the  left  hand 
thus,  ^,%,  is  written  ,5 ;  ^^^  ,45  ;  y^V  5^^^^  &^- 

But  if  tlie  numerator  has  not  so  many  places  as  the 
denominator  has  cyphers,  put  so  many  cyphers  before  it, 
viz.  at  the  left  hand,  as  wilPmake  up  the  defect  5  so  write 
y|^  thus,  ,05 ;  and  ^^^^  0  '^'^^^'^9  5^^^?  ^c. 

Note.     The  point  prefixed  is  called  the  separatrix. 

Decimals  arc  counted  from  the  left  towards  the  right 
hand,  and  each  figure  takes  its  value  by  its  distance  from 
the  unit's  place  ;  if  it  be  in  the  first  place  after  units,  (or 
separating  point)  it  signifies  tenths  5  if  in  the  second, 
hundredths,  &c.  decreasing  in  each  place  in  a  tenfold  pro- 
poj'tiou,  as  in  the  following 

NUMERATION   TABLE. 


Ui    (A 


7654521  S$456r 

jr/4oJc  JS'^iimhers.  Dccimai-s 


(>  jpluu-s  placed  at  the  right  hand  of  a  decimal  fracumi 
do  not  alter  its. value,  since  every  si-giuficant  figure  con- 
fv:.» -i".  :  1  ::r...^  the  same  place:  so  ,5  ,50  and  ,500 are 
4/  %  and  equal  to  ^^  or  ^. 

^  j^'iaced  at  the  left  hand  of  decimals,  de- 
crease t  eir  value  in  a  tenfold  proportion,  bj  removing 
them  Irirtoer  from  the  decimal  point.  Thus,  ,5  ,05  ,005, 
ikf.  are  five  tenth  parts,  five  hundredth  parts,  live  thou- 
san^t^  parts,  &c.  respcctiveij.  It  is  tiicrefore  evident 
that  the  magnitude  of  a  decimal  fiaction,  compared  Vvith 
anofeer,  does  not  depend  upon  the  number  of  its  fi<jurej?, 
but  upe»  tlio  value  of  its  first  left  hand  figure :  tor  in- 
staMte,  a  fraction  beginning  with  any  figure  less  tlian  ,9 
su«ii  as  ,899229,  &c.  if  extended  to  an  infinite  number 
of  figures,  will  not  equal  ,9. 


ADDITION  OF  DECIMALS. 
RULE. 

1.  Place  the  mumliers,  whether  mixed  or  pure  decimal 
under  each  other,  according  to  tlie  value  of  their  places 

2.  Find  their  sum  as  in  v/hoic  numbers,  and  point  oiv 
so  many  places  for  the  decimals,  as  are  equal  to  the  great- 
est number  of  decimal  parts  in  any  of  the  given  numbers. 

EXAMPLES. 

1.  Find  the  sum  of  41,653+56505+24,009  +  1,6. 

"41,653       . 

Thus    J2^'05 
1,6 


iSum^  103,312  v»hich  is  105  integers,  and  -^-^^^  parts  of 
in  unit.  Or,  it  is  103  units,  and  3  tenth  parts,  1  hun- 
dredth, part,  and  2  thousandth  parts  of  an  unit,  or  1. 

Hence  we  may  observe,  fnat  Decimals,  and  Federai^ 
Money,  aze  sulkct  to  one,  and  the  same  law  of  notation^ 
$ij^  conse.ffiently  of  operation. 


,)%QlM.\;j    I'll  ACTIONS. 


(2) 

Yards, 

(3) 

Ounces, 

46,23456 

H,3456 

24,90400 

7,891 

17,00411 

2,34 

3,01111 

5.6 

For  since  dollar  is  the  money  unit;  and  a  dime  being 
tlie  tenth,  a  cent  the  hundredth,  and  a  mill  the  thousandta 
part  of  a  dollar,  or  unit,  it  is  evident  that  any  number  of 
dollars,  dimes,  cents  and  mills,  is  simply  the  expression 
of  dollars,  and  decimal  parts  of  a  dollar :  Thus,  11  dollars^ 
6  dimes,  5  cents, =11,G5  or  ll-jV^  doL  &c. 

B,  Add  tlie  ft Uowiag  mixed  numbers  together. 

(4) 

Dollars, 

48,9103 

1,8191 

3,1030 

,7012 


3?.  x\dd  the  following  sums  of  Dollars  together,  vix» 
gl2,S4565-f7,891+2,S4+14,i-,0011 

Ans.  SS6,57775,  or  g36,  5di,  Tcts,  T-^-^-^mills. 
G.  Add  the  following  parts  of  an  acre  together,  viz. 
,75694-,25+,654+5l99 

Ans,  l,8599acrej. 
r.  Add  72,5-1-32,071 +2,1574+371,44-2,75 

Ms.  480,8784 
S.  Add  30,07+2130,71+59,4+3207,1 

Ms,  3497,28 
9.  Add  71,467+27,94+16,084+98,009+86,5 

Ms,  300 
IQ.  Add  57509+,0074+,69+,8408+,6109 

Ms,  2,9 

11.  Add  ,6  +  ,099+,37+,905+,026  Ms.  2 

12.  To  9,999999  add  one  millionth  part  of  an  uni^ 
and  the  sum  will  be  10. 

13.  Find  the  sum  of 

Twentv-five  hundredths,      -     -     -    -    - 
Tkree  hundred  and  sixty -five  tlio«isandtlvs^ 
Sixtenilu;  and  ninemiliionths,     -    -    -    . 


Mswer^  1,215009 


80  r^zcjy :-.':.  yxACTioj^s. 

SUBTRx\CTION  OF  DECIMALS. 

RULE. 

Place  the  numbers  according  to  their  value;  then  sub- 
tract as  in  ^vhole  numbers,  and  pokit  oft' the  decimals  as 
Ih  Addition. 

EXAMPLE'5^ 

Bollai'S,  Inches. 

1.  From  125.64  2.  From  14,674 

Take    95,58756  Take    5,91 


$.  From  761,8109  719,10009  27.15 

Take     18^9113  7,1  CI  i;51 679 


6.  From  480  take  245,0075  ./?«.s.  234,9925 

7.  From  2S6  dols.  take  ,549  dols.  Jlns.  8235,451 

8.  Fnmi  ,145  tske.  ,09684  Jm.  ,04816 

9.  From  ,2754  take  ,9.S71  .^«.«.  ,0385 
iO.  From  271  take  215,7  Jlrn^.  55S 

11.  From  270,2  tr>ke  75,4075  .^tf.s.   194,7925 

12.  From  107  takfi  ,0007  .ins,  106,9993 

13.  From  an  unit,  or  1,  s/abiract  the  millionth  part  ol 
itself.                      '          '  .itis.  ,99l>999 


>IULTI PLICATION  OF  DECDJALS. 

1.  Whether  tliey  he  ir.ixed  numbers^  or  pure  Jcclmals, 
jJace  ihefactorvS  raiil  nmitiply  themas  iji  whole  numbers. 

2,  Vu\i\t  oW  so  WAUiy  fjgui-es  from  iho.  prodTirt  as  tJioro 
aredecin.ai  ph-^ees  in  both  the  Tactorj; ;  and  if  there  be 
not  so  many  places  in  the  product,  supply  the  defect  by 
prefixing  (cyphers  to  iliQ.  left  hand. 


DECIMAL  FRAOTIONS, 


^ 


EXAMPLES. 

1.  Multiply  5,236  2.  Multiply  3,024 

by    ,008  hj    2,23 


Product     ,041888  6,74352 

5.  Multiply    25,238  by  12,17  307,14646 

4.  Multiply       2461  bj    :j29  130,1869 

5.  Multiply       7853  by  S,5  27485,5 

6.  Multiply  ,007853  by  ,035  ,000274855 

7.  Multiply        ,004  by  ,004  ,000016 

8.  What  cost  6,21  yards  of  clotli,  at  2  dola.  32  cents,  5 
mills,  per  yard  ?  Jns,  814,  4^.  Sc.  Bj\%-vu 

9.  Multiply  7,02  dollars,  by  5,27  dollars. 

Ans.  36,9954^0^5.  or  g56  99cfs.  5j%vu 

10.  Multiply  41  dels.  25  cts.  by  120  dollars. 

Ms.  84950 

11.  Multiply  3  dels.  45  cts.  by  16  cts. 

•5ws\  jg0,5520i=55ct9.  9>mills. 

12.  Multiply  65  cents,  by  ,09  or  9  cents. 

Ms.  80,0585  ==5ces.  Shmills. 

13.  Multiply  10  dols.  by   10  cts.  Ms.  gl 

14.  Multiply  341,45  dols.  by  ,007  or  7  mills. 

Ms.  82,39+ 

^  To  multiply  by  10,  100,  1000,  &c.  remove  the  separa^ 
ting  point  so  many  places  to  tlie  right  hand,  as  the  mul- 
tipfier  ha«  cyphers. 

r Multiplied  by  10,  makes  4,25 

So  ,425  < by  100,  makes  42,5 

I by  1000,        is  425, 

For  ,425X10  is  4,250,  &c. 

Ill  ^  mum 


DIVISION  OF  decimals: 

UUT.E. 

t.  TliC  [L;lac<\s  of  the  decimal  parts  of  the  divisor  an4 
Cjijotifnt  c<>^(nted  tc»;ctlHT.  nv1s^  nlv^r^vs  be  equal  to  those 


1^ 


l^SiOlMAI.  FRACTIONS^ 


m  tlie  dividend,  therefore  dfvide  as  in  "whole  numbegs, 
and  from  the  right  hand  of  the  quotient,  point  off  so  ma- 
ny places  for  decimals,  as  the  decimal  places  in  the  divi- 
dend exceed  those  in  the  divisor. 

2.  If  the  places  in  the  quotient  are  not  so  many  as  the 
rule  requires,  supply  the  defect  by  prefixing  cyphers  to 
tiie  left  hand  of  said  quotient. 

Note.  If  the  decimal  places  in  the  divisor  be  mare 
than  those  in  tlie  dividend,  annex  as  many  cyphers  to  the 
dividend  as  you  please,  so  as  to  make  it  equal,  (at  least) 
to  the  divisor.  Or,  if  there  be  a  remainder,  you  may 
annex  cyphers  to  it,  and  carry  on  the  quotient  to  any  de- 
gree of  exactness. 

EXAMl'LES. 


^351)77,41 14(8,14 

rem 


1,331 
951 


3804 
3804 


5,8),S1318(,05&1 
190 


00 


00 

♦ffnszrers. 


$.  Divide  780,517  by  24,5 

4.  Divide  4,18  by  ,i8TS 

5.  Divide  7.25406  by  957 

6.  Divide  ,00078759  by  y5'^l5 

7.  Divide  14  by  ^65 

8.  Divide  S!246,1476  by  ^604,25 

9.  Divide  §186513,239  bv  S304,81 
10.  Divide  81,^-8  hy  »8,3i 
M.  Divide  5Gci&.  by  1  del.  12ets. 
IS.  Divide  1  drtllar  by  1.^  cents. 
15.  y  iJlj  or  21,75  yards  of  c!o»th  cost  34^317  dollars^ 

what  will  one  yard  cost  ?  21,577 

Note.     When  decimate,  or  whole  numbers,  are  to  be 
Sivideil  bv  10,  ICO,  1000,  &c«  (vlr..  unit v  with  cyphers) 


S2,l£ 

,25068+ 

,00758 

.00150+ 

,058356+ 

,40736+ 

611,9+ 

,154+ 

8,533+ 


DEeiMAL  f  RAOTXOKS.  o^ 

a  isperfottned  by  removing  the  separatrix  in  the  divi- 
dend, so  many  places  towards  the  left  hand  as  there  a5C 
cjphers  in  the  divisor. 


.1 


'11 


EXAMPLES. 


'  10,  the  quotient,  is  ^7^9. 

57SL  divided  hj{  100,         -  ^    -         5,72 

1000,       -        -  ,572 


REDUCTION  OF  DECIMALS. 

CASE  I. 

To  reduce  a  Vulgar  Fraction  to  its  equivalent  Decimal 
RULE. 

Annex  cyphers  to  the  numerator,  and  divide  by  the 
denominator ;  and  the  quotient  will  be  the  decimal  re» 
quired. 

Note.  So  many  cypher^  as  you  annex  to  the  given 
jitmerator,  so  many  places  must  be  pointed  in  Hh^  quo- 
tient; and  if  there  be  not  so  many  places  of  figures  ia 
the  quotient,  make  up  the  deficiency  by  placing  cyphers 
to  the  left  hand  of  the  said  quotient^ 

EXAMri.ES. 

1.  Reduce  -f  to  a  decimal.  8)1,000 

JUns,     ,125 

2.  "What  decimal  is  equal  to  i  ?        Answers.  ,5 
S.  What  decimal  is  equal  to    i?       -    -    ^    -      ,75 

4.  Reduce  4-  to  a  decimal.      - ,2 

5.  Reduce -54  toadccima],      -     -    -     -    „        ,6875 

6.  Reduce  |^J  to  a  decimal.      ^     -     -     -     ..     -      ^85 

7.  Bring //to  a  decimaL    ------      ,09375 

8.  What  decimal  is  equal  to  -^^  ?       .     .     ,0S7037-r 

9.  Reduce  ^  to  a  de<;imal.     '*-    -     ^    -     ,333i5C>3-f 
1!K  R«duce  ^^j^i  to  its  equivalent  decimal.     -    ^OOS 

*    Itr Reduce-^ to «  decimal.     -    -    -       j- 523075^ 


84 


DECIMAL   FRACTION':. 


CASE  11.  1 

Vo  reduce  quantities  of  several  denominations  to  a 
Decimal, 

RULE. 

Bring  the  given  denominations  first  to  a  vulgar  fraction 
bj  Froblem  HI.  page  76 ;  and  reduce  said  vulgar  frac- 
tion to  its  equivalent  decimal ;  or 

Rule  2.    Place  the  several  denominations  above  each 
other,  letting  the  highest  denomination  stand  at  the  hot 
torn ;  then  divide  each  denomination  (beginning  at  tht 
top)  by  its  value  in  the  next  denomination,  the  last  qUD* 
tlent  will  give  the  decimal  required. 

EXAMPLES. 

1.  Reduce  12s.  6d.  Sqrs.  to  the  decimal  of  a  poun^* 
12 


150 
4 

9G0)6O5,00O000(, 

5760 

628125 

Jlnswer. 

sroo 

1920 

\ 

Rule  2. 

7800 
7680 

12 
20 

6^75 

12,5625 

1200 

9G0 

£400 
1920 

,628125 
6 

4800 
4800 

DECIMAL  MACTION?^  88^   > 

5.  Ileducfe  15s.  9(1.  Sqrs.  to  the  decimal  of  a  pound* 

Ans.  ,790625 

5.  Reduce  9d.  Sqrs.  to  the  decimal  of  a  shilling.  , 

Jlns.  ,8125      ^ 
4.  Reduce  3  farthings  to  the  decimal  of  a  shilling. 

Ms.  ,0625 
^  5.  Reduce  3s.  4d.  New-England  Currency,  to  the  de 
cimal  of  a  dollar.  Ms,  ^555555+ 

6.  Reduce  12s.  to  the  decimal  of  a  pound,    dns.  ,6 

Note.  When  the  shillings  are  even,  half  the  number 
ivith  a  point  prefixed,  is  their  decimal  expression ;  but 
if  the  number  be  odd,  annex  a  cypher  to  the  shillings,  and 
then  by  halving  them,  you  will  have  their  decimal  ex- 
pression. 

7.  Reduce  1,  2,  4,  9,  16  and  19  shillings  to  decnnals 
Shillings    1        2        4        9        16        19 

Mswers.  ,05        ,1       ,2     ,45         ,8       ,95 

8.  What  is  tl\e  decimal  expression  of  4L  19s.  6^d.? 

Ms,  £4,97708+ 

9.  Bring   34^.  iGs.  73d.  into  a  decimal  expression. 

Ms.  £34,8322916+ 
10*  Reduce  25Z.  19s.  ojd.  to  a  decima-iT 

Ms.  £25,972916+ 

11.  Reduce  Sqrs.  2na.  to  tlie  decimal  of  a  yard. 

Ms.  ,875 

12.  Reduce  1  gallonjo  the  decimal  of  a  hogshead. 

Ms.  ,015873+ 

13.  Reduce  7oz.  19pwt.  to  the  decimai-cf  a  lb.  troy. 

Ms.  ,6625 

14.  Reduce  Sqrs.  211b.  Avoirti^pois,  to  the  decimal  of 
an  cwt.  .  Ms.  ,9375 

15.  Reduce  2  ruodiS,  16  perciKis  to  the  decimal  of  an 
acre.  ^  *  Ms.  ,6 

16.  IL'duce  2  fccjt  6  inches  to  the  decimal  of  a  yard. 

Ms.  ,833333+ 

17.  Reduce  5fiir,  -iGpo.  to  the  decimal  of  a  mile. 

*J?ts..,675 

18.  Reduce  4i  calendai'  mo*ntlig  to  ilie  decimal  oi' 


O  3£G1MAL  hhaotiqki 

CASE  in. 

Ti^find  the  value  of  a  tfecimal  in  tHe  known  parts  of  thi 

integer. 

R¥LE. 

1.  Multiply  the  decimal  by  the  number  of  parts  in  tim 
Wxt  less  denomination,  and  cut  off  so  many  places  for  a 
remainder,  to  the  right  hand,  as  there  are  places  in  the 
given  decimal. 

2.  Multiply  the  remainder  by  the  next  inferior  denom* 
illation,  and  cut  off  a  remainder  as  before  5  and  so  on 
tlnough  all  the  parts  of  the  integer, and  the  several  de- 
nominations standing  on  the  left  hand,  make  tlie  ai>sweif. 

EXAMPLES. 

1.  What  is  the  vake  of  ,5724  of  a  pound  Sterling? 
£.  ,57»k 
20 


11,4480 
12 

5,.sr6o 

4 

1,5040  Jlns.  Uss.  5d.  t,5^. 

2.  What  is  the  value  of  ,75  of  a  pound  ?     Ans.  155. 
S.  What  is  the  valie  of  ,85251  of  a  pound  ? 

Ms.  17s.  Od.  2,4grs. 

4.  What  is  the  value  of  ,040625  of  a  pound  ? 

Ms.  9ii. 

5.  Find  the  value  of  ,8125  of  a  shilling.     Ans.  O^d. 

6.  What  is  the  value  of  ,617  of  an  cwt. 

Jns.  ^qrs,  ISlb.  loz.  lOfidr. 
T.  Find  the  value  of  ,76442  of  a  pound  troy. 

^^tm.  9d».  Sipiet.  ll^r. 
S.  Whatis  tlic  value  of  .875  of  a  yd.  ?    Ms.  Sqrs.  ^l 
9.  Wktt  i#  the  value  of  ,875  of  a  hhd.  of  \^ne  ? 

M%  £5>^(d.  0(yt  Iff,. 


1! 


DEC  J  MAT.  FRACTIONS. 


iT 


10.  Find  tlie  proper  quantity  of  ,089  QJ||Ji:ule, 

Ans.  kSpo.  P.ijds^t.  llj04i}K 

11.  Find  the  proper  quantity  of  ,9075  of  an  acre. 

Ana,  Sr.^25,2po. 

12.  What  is  the  value  of  ,569  of  a  year  of  S65  dajs  ? 

Ans,  20rrf.^l6/i.  267/1.  24sec. 
15.  What  is  the  prefer  quantity  of  ,002084  of  a  pound 
^roy?  Ans.  12,003S4^r, 

14.  What  is  the  value  of  ,046875  of  a  pound  avoirdu- 
a\s  ?  Ans.  12rfr. 

15.  What  is  the  value  of  ,712  of  a  furlong? 

Jns,  mvo.Qyd,  Iff.  11.04f?!. 

16.  What  is  the  proper  quantity  of  ,142465  of  a  year  ? 

\:hi^,  51,999r25ifff2/s- 


CONTRACTIONS  IN  DECIMALS. 

PROBLEISI  I. 

A  CONCISE  and  easy  method  to  find  the  decimal  6f 
any  number  of  shillings,  pence  and  farthings,  (to  three 
pinces)  by  Ixspkction. 

RULE 

1.  Write  half  the  greatest  even  number  of  shillings  for 
the  first  decimal  figure. 

2.  Let  the  farthings  in  the  given  pence  and  fartliings 
possess  the  second  and  third  places ;  observing  to  increase 
the  second  place  or  place  of  hundredths,  by  5  if  the  shilr 
lings  be  odd  5  and  the  third  place  by  i  \vhen  the  far- 
'.hings  exceed  12,  and  by  2  when  they  excmi^^il  36, 

EXAMPLES. 

I.  Find  the  detimal  of  Ts.  9  Jd.  by  inspection. 

,3     ==:i  6s. 
5      for  tlie  odd  shillings. 
39=the  farthings  in  9|d. 
2     for  the  cxce^ii  of  Sti. 

£.  ,391— dtxima)  required 


r 

dg  DECIMAL  FRACTIONS* 


5.  Find  tlie  decimal  expression  of  16s.  4 id.  and  17   £ 
*^^'        .A  .fws.£.  ,819,  flntfjQ.  ,885     I 

S.  WritePPown  £47  18  10^  in  a  decimal  expression* 

\  .         Jns,  £47,943 

4.  Red^ice  £  1  8s.  2d.  to  an  equivalent  decimal. 

Ans.  £1,408 

PROBLEM  II. 

A  short  and  easy  method  to  find  the  value  of  any  deci- 
mal of  a  pound  by  inspection. 

RULE. 

Double  the  first  figure,  or  place  of  tenths,  for  shillings^ 
and  if  the  secoud  figure  be  5,  or  more  than  5,  reckon 
another  shilling  ;  then,  after  t]\\s  5  is  deducted,  call  the 
figures  in  the  second  and  third  places  so  many  farthings, 
abating  1  when  they  are  above  12,  and  2  when  above  36^ 
and  the  result  will  be  the  answer, 

Note.  When  the  decimal  lias  but  2  figures,  if  any 
filing  remains  after  the  shillings  are  taken  ou^  a  cypher 
must  be  annexed  to  the  left  hand,  or  suppose  i  .  u  a    ;  r. 

EXAMPLES. 

1.  Find  the  value  of  £.  ,679,  by  inspection. 
12s.=doubleof  6 
1       for  t!.c  5  in  the  second  place  which  is  to 
[be  deducted  out  of  T. 
^d  rid.=29  farthings  remain  to  be  added. 

Deduct  id.  for  the  excess  of  12. 


1 


j3ns.  i3s.  rd, 

%  Find  the  value  of  £.  ,876  by  inspection. 

Jins.  17s.  6id, 
S,  Find  the  value  of  £.  ,842  by  inspection. 

Ans.  16s.  lOd. 
4«  Find  the  value  of  £.  ,097  by  inspection. 

dn$.  is.  IIK 


JS^ii'-York,  and  ') 
JS^orth' Carolina.  5 


REDUCTION  OF   CURRKNCIES. 

RULES, 

1^  OH  reducing  the  Currencies  of  the  several   United 
States*  int'j  Feilcral  ."^toiej. 

CA.8E  I. 
To  reduce  the  currencies  of  the  different  states,  where 
&  dollar  is  an  even  number  of  shillings,  to  Federal  Money. 
They  are 
(A"e  ic  'Engla  nd^ 
J  Virginia^ 
^\  KentkckVy  and 

L'"        •■■- 

!.  Vv'hcii  liic  sum  consists  of  pounds  only,  annex  a  cy 
|;her  to  the  p'vjnds,  vjmI  divide  by  i\alf  the  number  of 
5;ijiHiiigs  in  a  dollar  5  tiie  quotient  will  be  dollars.t 

'2,  But  if  Hie  sura  consists  of  pounds,  shiliings,  pence, 
f'X.  bring  ciie  given  sum  into  shillings-  and  reduce  the 
pence  and  farthing:s  to  a  decimal  of  a  shilling ;  annex  said 
decimal  to  theshilfin^s,  Avith  a  decimal  pointbetvveen,  tlieu 
divide  the  v.h.de  by  the  number  of  shillings  contained  in 
:i  dollar,  and  tiie  quotient  will  be  dollars,  cents,  mills,  ^c. 

•^'Formerly  the  pound  was  of  the  same  ster!i>-<;  vakie  in  all 
the  colonics  as  in  Groat-Britain,  and  i  Spanish  Dollar  worth 
4sf>— but  the  l^^gisiatures  of  the  different  colonies  emitted  bills 
of  credit,  v/hich  afterwards  depreciated  in  their  value,  ki 
s,ome  states  more,  in  others  less,  he. 

Tlius  a  dollar  is  reckoned  in 


Js^W' Engl  and  r\ 
Virginia  y  \n 

Kentucky,  and  f 

.VeW'Fork,  ^'  " 
«V*.  Carolina. 


8s 


JVew-rkrseij^ 
Pennsylvania^ 
Bdaware*  and 

Mary  land . 


>7s6 


South'   "^ 
Carolina,  \.^ 
and       p^^ 
Georgia.  J 


f  Adding  a  cypher  to  the  pounds,  multipues  the  whole  by 
•O,  bringing  them  into  tenths  of  a  pound  ;  then  because  a 
iQi'-r  is  just  three^tenlhs  of  a  pound  N.  E.  currency,  divi- 
s!),;  tbos^f  tenths  hy  S,  brings  them  into  dollars,  fcc.    See 

a*  & 


I .  n *M i  I c«  - 1 ' i\9,l and  and  Virginia  Cii rrency . 

;o  '¥<<':■-.  -,  5)730 


2.  Reduce  45/.  I5s.  7hd,  New-England  currencj,  to 
20  [federal  money. 

AdoI]ar=6)915,e25  .^s-^  -.. 


Note.  1  farthing  is  .23  "1  which  aniicx  to  tac  pence, 

2  — -     .T^    ,50  1. and  divide  bj  12,  you  wftl 

3  __     —^    .75  J  luwcllr^.  declnial  required. 
5.  Reduce  S45^.  10^.  1  U J.  Ncw-IIainpshirit,  &c.  cur 

rencj,  to  Spanisli  ndlled  dollai-s,  oj*  federajl  monev. 
£545     10     111  .'         ^ 

20  .-  cl  - 

12)1152500  ■ 


6)691059375 


^92  7 '^  clecimu. 


81151,8229+  ^fis, 
4.  Reduce  105^  lis.  S|(i.  New-York  and Nd-th-Cai'd* 
Ihia  currency,  to  federal  money. 

/:i05     14     SJ         "  d. 

^)3,.: 


20  12)3,7500 


A  dollar=B)21 14,3125  ,3125  decimat 

^264,289  05  Ans. 
Or  S  dcm.  ^^^ 
5.  Reduce  43 IL  New- York  currency  to  federal  raoneyt 
This  being  pounds  only.*—.    4)4310*^ 

Ans.  81077^=1077550 


*j2  aollar  is  Ss.  in  tkts  currency — ^4=^\  of  a  pound  $ 
Hin'('ff}r:^  ■  '.■<'  /.»^  hij  10,  and  divide  %  4,  brivgs  tlm 
pounds  ■  /F,  §-c. 


^  REDUGTIOK   OF   CURKEKCIES.  0^ 

6.  Reduce  28^  lis.  6d.  New -England  and  Virgiiii^ 
C-iiTCucy,  to  federal  money.  Ans,  g95j  25cts. 

r.  Change  46SL  10s.  Sd.  New-England,  &c.  currency, 
fi)  federal  moTiey.  Jlns.  S1545,  lids.  Im.-J-' 

o.  Reduce  3f5L  193.  Virginia,  &c.  currency,  to  federal 

ney.  Jlns.  ^119,  8Scfs.  S?u.+ 

9.  Reduce  214/.  10s.  7 id.  New-York,  &c.  currency, 
to  icderal  money.  Ms.  g536,  Z2cis.  8m.-i- 

10.  Reduce  304^  lis.  5d.  North-Carolina,  &c.  cur- 
cy,  to  federal  money.        Arts.  grGl,  42cf5.  Tm.-^- 

:f.  Change  219/.  lis.  Tjd.  New-England  and  Vir- 
ginia currency,  to  federal  money.      Ans.  g7Sl,  94c^s.-i- 

12.  Change  24iZ.  Ncv/-En^land.  occ.  currency,  int© 
federal  money.  Jjis.  S803,  SScts.-f 

1?.  Erin::  20!.  13b.  v-England  currency,  iuto^^ 

dollars'.  Jhis.  ,§69,  74cts,  6^m.4- 

14.  Redi  _re-A-york  currencvjto  federal  mo- 
l^-^y:  "    Jins.  Sliro 

15.  Reduce  ITs.  G^d.  New-York,  ike.  currency,  to 
«;l!-\rs,  £xc.  *^/:s.  £2,  .2-acfi?.  6j5m>-{- 

15.  Borrowed  10  English  crowns,  at  6s.  Sd.  each,  how 
?:^y  doilr.rs  at  Gs.  eacli,  will  pay  the  debt ; 

Ms.  Sll, 
Note. — Tliere  are  several  short  practical  methods  of 
lucing  New-England  and   New -York  cun^encics  to 
Federal  Monev,  fof  wIii(Bfcee  the  Appendix. 

To  rcd'jce  the  currency  of  New-Jersey,  Pennsylraul^i^ 
IKdawareand  Ivlarvland,  to  Federal  Money. 
RULE. 
'^"''I'inly  the  given  sum  by  8,  and  divide  iSid.  product 
:'l  t!ie  quotient  will  be  dollars,  &c.* 

JiXAMPLES. 

:.  Reduce  24dZ,  New-Jersey,  Sec.  currency,  to  federal 

<8==::1960,  ttjz^Z   1960-4-5  =:S65?4  =S653,  53Jrf^. 
«:v  i  ff.. — .When  there  are  shillings,  pence,  &c.  in  the 

/!  ('-''aris  7s.6d.=^90d.in  this  cur renctj^zT^^^j^^:^  of 
%'^:r'  ? ;  therefore,  r.mltipljiug  by  8,  and  dividing  hy  3> 


^^  n.  nUGTIOX    OF   CJsURRENCIES. 

« 

given  sum,  reduce  them  to  the  decimal  of  a  pound,  tlien 
multiply  and  divide  as  above,  &c. 

2.  Reduce  36/.  lis.  S^d.  New-Jersej,  &c.  currencj, 
to  federal  money.     £36,5854  decimal  vahie. 

8 

g 

3)292,6852(97,50106  Ans,     answef.s, 
£.      s.    (L  g     c^s.  711. 

5.  Reduce  240    0    0  to  federal  money  640  GO 

4.  Reduce  1-25    8    0    '     —  S34  40 

5.  Reduce     99    7    6:^      ^^--  265  00  5    -f 

6.  Reduce  100    0    0        -~^.  266  66  6   -f 

7.  Reduce    25    3    7        67  14  4 

S.  Reduce      0  17    9        2  56  6,6 

C^SE  ill. 

To  reduce  the  currency  of  South-Carolina  and  Georgia, 
to  federal  money. 

RULE.  _  ^      .      •■ 

Multiply  the  given  sum  by  30;?  and  divide  the  product 
by  7,  the  quotient  ^^i\\  be  the  dollars,  cents,  &c.* 

EXAMPLES. 

1.  Reduce  lOOZ.  South-Carolina  and  Georgia  currency, 
to  federal  money. 

100/.x50=S000  5  5000-v-7=g428.5714  Ms. 

2.  Reduce  54/<  1 6s.  9  jd.  Georgia  currency,  to  federal 
money,  54,8406  deciwdl  Expression  * 

30 


7)1645,2180 


dnS.  255,0311  ANSWERS. 

£.  s,    d.  S  cts,  m. 

S:  Reduce  94  14  8  to  federal  money,  405  99  S-f 

4.  Reduce  19  17  G^    85  18  7-^ 

5.  Reduce  417  14  6    1790  25 

6.  Reduce  140  10  0    602  14  2+ 

7.  Reduce  160  0  0    685  71  4 

— k -^-, ■ — - 

*4s.  Sd.  or  56d.  to  the  efo/tar=^2^V=/7  ^/  *  ]?omd; 
tfiex^ore  x30-r-7. 


REDUCKION   ©F    COIN. 


to- 

s. 

d. 

lled  lice    0 

11. 

G 

Ileduce  41 

ir 

9 

^  1 


55  ct^'  m. ' 
2  46  4«h 
179  51  4f^ 


CASE  IV. 

I'l;  reduce  the  currency  of  Canada  and  Nova-Scotia^  to- 
Federal  Money. 
RULE. 
Multiply  tlie  given  sum  by  4,  the  product  will  be  dollars.. 
i>oTE.  Five  shillings  of  this  currency  are  equal  to  a 
dollar ;  consequently  4  doilais  make  one  pound 

EXATviPLES. 

1.  Heduce  125?.  Canada  and  Nova -Scotia  cUrrenc}',  ID* 
^C'lciT,!  innnoy.  1C.5 

Ms.  B500 

2.  Reduce  551.  10: .  6d.  Nova-Scotia  cuirency,  toCSlr 
lArs.  55,59.5  decimal  value. 

^'      4 

S  cis. 

Ms.  82225  100=222  10  AKSWERST^ 

5.  Reduce  241  18    9  to  federal  money*  96r  75 

4.  Reduce    58  13    6i            234  70 

5.  Reduce  528  17     8              « 2115  5S 

6.  Reduce      12    6             4  50 

7.  Reduce  224  19    0              -    899  80 

8.  Reduce      0  13  lU            2  79 


REDUCTION  OF  COIN. 

RULES 

For  reducing  the  Federal  Money  io  the  currencies  of  thft 

several  United  States. 

To  i?educe  Federal  Money  to  the  currency  of        i^^. 

„    ^^S^'"'^^   ^rM«ltiplythegivensumby,5 
1  <  AT.„fc.;'i-,;.  nr,Ac-i\  ar4t!.eproductmllbepoui\a* 


1 


/v-en^'cfc^  nnd\i\  and  the  product  mil  be  pouiid* 
Tenwssit       j   «  Und  decimals  ot  a  pound.. 


^,Gj  ik-c* 


ULES)  f6r  redyi^n^  tlio:  currencies  of  tlie 


aUt' 


to  the  par  of  a^pbe  others.     See  tl 

ji'jglit  hand,  till  jou  coir*e  under  the  required  cuiTencj: 


e\M5fal  {jnU 
vcn  curiiency 


Virgirda, 
Kentucky^ 

and 
Tennessee, 


^\  England. 

Virginia^ 

Kentucky  ^^ 

and 
*Fe.nnessep.. 

^'iw-Jersey, 
(Pennsylvania^ 
j   Delaware^ 
\        and 
j  Maryland, 


WeW'Forkf 

\^orih'CarO' 

►       linn,       X 

IS'Quth'Ccfroli- 

na^  and 
Genr2:'a, 


JSI'eW'Je^^sey^ 
Fennsylvafiia^ 

Delaware^ 
and 

Maryland, 


Deduct  one, 
liiVii  from  the 
jxiven  sum. 


V 


Deduct  one 
4th  from  the 
New-York, 

&c. 


^ew-Yorkj 

'  and 
JVl  Cnralimz. 


Add  pne  4th 
to  the   given 

sum. 


Add  one  Sc 
to  the  give: 

iiUUl. 


Deduct  one 
ifith  trom  the 

New-York. 


u;iven  sum  Ov 
9,  and  divide 
^]i(^     product 


I  Multiply  the. 
given  sum  by 
45,  and  divide 
the  product 

IVy  28. 


Add  one  5th 
I  to  the  Cana- 


Add  one  half 
G  the  Canada 


Add  one  fif- 
teenth to  the 
'i:iven  sum. 


IMuitipiy  the 
ffiven  sum  bv 
12,  and  di- 
vide the  pro^ 

duct  by  T. 


Multiply  the 
given  sum  bj 
8,  and  divid 
tlie     product 
by  5.  ^ 


iultiply the   Multiply  tht 


;iish  ^um  said 

;oue  thiri 


.^D^l'hnior^ 
^y^^>j  and  m- 
j'vide  thii  pro- 
iduct  by  3 


suns 


English 

b>''  "l6,anti  r.'j-. 


ih- 


t;ie  ., 


llULf^il  FOIl  HEDUfiXJiiS,  &.C^ 


.1\ 


ted  States,  also  Canada,  Nova -Scotia,  and  Sterling,  eatli 
in  the  left  hand  column,  and  then  cast  your  eye  to  the 
and  you  will  have  the  rule. 


South-Carolina^ 

and 

Gior^lrt, 


'     Canada^ 

and 
JVova-Scotia^ 


Sterling* 


Multiply  the  giv- 
en sum  by  7,  and 
divide  the  product 
Dy  9. 


rSJultlpIy  iii\Q  div- 
ert sum  by  28,  ^xn^l 
divide  the  product 


Multiply  the  giv- 

II  Sum  by  7,  and 
ivide  the  product 
by  12. 


Multiply  t^'i^fi  giv 
en  sum  by  5,  and 
divide  the  product 
bv6. 


Deduct  one  third 
rom  the  given 
sum. 


Multiply  the  giv- 
en sum  \ij>  5.  and 
divide  tlic  pre  duct 
by  8.  ' 


Deduct 
fourth  from 
jdven  sum. 


one 
the 


Multiply  the  giv- 
en sum  by  3,  and 
divide  the  produQ^ 
by  5. 


Multiply  the  giv^ 
en  sum  by  9,  and 
divide  the  producJ 
Ibv  16. 


1  Multiply  the  giv- 
Jen  sum  by  15,  and 
[divide  the  product 
joy  14. 


I 


From  the  glvenj 
sum,   deduct   cnri 

L\,en-y-cip;]ua.      ! 


I  Deduct  one  flF-JI 
jUenth  irom  the.; 
jgiv'Bn  sum. 

I J ' 

\    T(i  tiie  EngUsii 

Fiiey    add 
eitty'Seventli. 


Add  one  nintu 
onclto  tiio  givwi  Slim. 


'  lieuuci 

tenth     from 
i^iven  iTwim, 


thfil 


vr, 


98  HEDUCTIOX   OF    CQIN. 

APPLICATION 

Of  the  Rules  contained  in  the  foregoing  TahU. 

EXAMPLES. 

I.  Reduce  46L  10s.  6 J.  of  the  currency  of  New-Haiflj» 
sliire^int©  that  of  New- Jersey,  Pennsylvania, &c. 

See  i\\Q,  Rule 
in  the  Table. 

Jlns.  £58     3  1^ 
£.  Reduce  25/.    13s.   Ot^.  Connecticut   currency,  to 
N«w-York  currency. 

£.  s.  d. 

$)25  13  9 

By  the  Table,+}  &c.  -f  8  H  S 


4)f6 
+11 

10  6 

12  7J 

\ 


^^  ^ns.  £34    5  0 

3.  Reduce  125/.  10^.  Ad.  New-York,  &c.  turrcncy, 
South-Carolina  currency. 

£.  5.  d. 

Rule  by  the  Table,  125  10  4 

xr,-f-by  12,  &8,  7 

12)8r8  12  4 

Jins.  £73    4  4i 

4.  Reduce  46f.  lis.  8i.  New-York  and  North«Car<K 
jVia  currency,  to  sterling  or  English  Money. 

£.  5.  iZ. 

AG  n  8 
9 


See  the  Table.    )       16=4x4Ml9    5  0 
Xgiven  sum  by  v  4)104  16  3 

9n-16,&c.         J  

4ks.  ^26    4  Oj 


RE'DUCTiON    or    COIN.  ^iL 

To  rexluce  any  of  thf  diflerent  currencies  of  thc^scrW 
ral  States  into  each  other,  at  par ;  yoii  may  consult  the 
preceding  Table,  which  will  pve  you  the  fiules. 

MORE    EXAMPLES   FOP*.   EXERCISE* 

5.  Reduce  84^  10s.  M.  New-TTa/r»r.<^iiIre,  &.c.  curren-^ 
cy,  intoNew-Jeisev  curreiicy. 

,  .  J-  £105  ISs.  4iL 

6.  Reduce   120/.  8>\  ScL   Connecticut  currency,  into 
New-Y«rk  currency.  Jns,  £160  lU.  (>/. 

7.  Reduce    120L    lOs.   ^?assachasetis  currency,  into 
South-Carolina  and  Georgia  currency. 

Jns,  £93  Kfr  5iiL 

8.  Reduce  410L  I8s.  Ih'/.  Rhode-Island  currency,  in- 
to Canada  and  Nova-Scotia  currency. 

Jns,  £542  9s.  id 

9.  Reduce  524^.  8s.  4d.  Virginia,  5tc.  currency,  into^ 
Sterling  money.  *      .  Jus,  £39o  Gsi  Sd,     ^ 

10.  Reduce  214/.  9s.  2f?.  New-Jersey,  &c.  currency,  T 
fauto  New-Hampshire,  Massachusettji,  ^c,  currency.     '   ^ 

.ins,  £171   lis.  4d.     |J 

11.  Reduce  lOOL  New-Jersey,  ^cc.  currency,  into  N.^' 
York  and  North-Carolina  currency.  * 

'M^.  £105  ISs.  4 J.      • 
i2.  Reduce   100/.  Delaware  an4  ^lajylar.d  currency^ 
in,to  Sterling  monev,  »J^>5.  ;(J60. 

13.  "Reduce  116?.  -York curreii(?y,  into  Con* 
n€cticut  currency.  .Ins.  £3V.  7s.  Cd,     M 

14.  Reduce    i  l£/.   Ts.    Sd,  S.  Cnrolhia  a!id  Georgia 
currency,  into  Connecticut,  &c.  currency.  w^ 

M$/£IU  93.  $id. 

15.  Reduce  100/.  Canada  and  Nova-Scotia  currency, 
into  Connecticut  currency.  *dns,  £   I-^^IO, 

16.  Reduce  116/.  14-^,  9d,  Sterling  money,  into  Con- 
necticut currency.  Jins,  £155  ISs, 

17.  Reduce   104/.  IGs.  Canada  and  Nova-Scotia  cur- 
rency, into  New- York  cirrency.  ^ns,  £167  4s. 

18.  Reduce   100/.  Nova-Scotia   ciinency,  into  Nejv- 
Je;;3ey,  &c.  currency.  '  Jhis,  £150 


^f  ..    Ky.LK   OF   XHRE.^    DIRECT. 

RULE  OF  THREE  DIRECT. 

,  X  HE  Rule  of  Tiiree  Direct  Teaches,  by  having  Hwe.% 
Tuimbers  given  to  find  a  fourth,  which  shall  have  the  same 

^proportion  to  the  third,  as  tlie  second  has  to  the  first. 
1.  Observe  that  two  of  the  given  liimbers  in  your 
jl^question  are  always  of  tlie  sair^e  name,  or  kind ;  one  of 
Oxvhich  must  be  the*^  first  number  in  stating,  and  the  other 
,  the  third  number ;  consequently,  the  first  and  third  num- 
f^  bers  must  always  be  of  the  same  name,  or  kind  ;  and  the 
^  other  numbBr,  which  is  of  the  same  kind  with  the  answer, 
r  or  tiling  For..!>;]it,  will  always  possess  the  second  or  middle 

\  2.  lh%tly''(l  term  is  a  demand  ;  and  may  be  known  by 
these  or  the  like  v/ords  before  it,  viz.  What  will  5  What 
co^  ?  How  many  ?  } Jo w  far  ?  Kow  long  ?  or,  Ho w 
much  ?  &c. 

RULE. 

1.  State  i]iQ  question ;  that  is,  place  the  numbers  s« 
tiiatthe  first  and  third  terms  may  be  of  the  same  kind; 
and  the  second  term  of  the  same  kind  with  the  answer,  or 
thing  sought. 

2.  Bring  tlie  first  and  tiiird  terms  to  the  same  denom- 
sjination,  and  reduce  the  second  term  to  the  lowest  name 
sT  mentioned  in  it. 

\  S.  Multiply  the  second  and  third  terms  togetlier,  and 
\flivide  their  product  by  the  first  term  5  the  quotient  will 
J  be  the  answer  to  the  question,  in  the  same  denomination 
^you  left  the  second  term  in,  wliich  may  be  brought  into 
y  any  other  denomination  required. 
\      The  metliod  of  proof  is  by  inverting  the  question. 

NOTE.- -The  following  methods  of  operation,  when  they 
can  bo  used,  perform  the  work  in  a  much  shorter  manner 
than  the  general  rule. 

1.  Divide  the  second  term  bj^  the  first;  multiply  the  quo- 
tient into  the  third,  and  the  product  will  be  the  answer.     Or 

2.  Divide  tlic  third  term  by  the  first ;  multiply  the  quotient 
i;ifo  tlie  second,  and  the  product  will  be  the  answer.     Or 

3.  Divide  the  first  term  by  the  second,  and  the  third  oy 
that  quotieiit,  and  the  last  quotient  will  be  the  answer.     Or 

4.  Divide  tJie  first  term  by  tae  third,  and  the  second  hj 
tRr,t  ffUCtiRnt,  and  th^  Irtst  q\tdti6nt  will  b<;  the  answer.^ 


rt:t.k  or  TH^I^  upiRxc  '.'. 


U):. 


>Ev^MPL^s, 


t.  If  6  yards  of  cloth  coSfQVlollarsiHvliat  ivlU  iS^ yards 


cost  at  the  same  rate  ? 

Fds.  S     Yds. 

Here  20  yards,  whiclx  moves 

6  :  9  :  :  20 

the  question,  is  tlie  third  term  5 

9 

6  yds.  the  same  kind,  is 

the  first. 

and  9  dollars  tlie  second. 

6)180 

Ms.  gSO 

2.  If  20  yards  cost  5 

0  dels.      5. 

If  9  dollars  will  buy  6 

what  cost  6  yards  ? 

yards 

;,  how  many  yards  will 

rds.    S      Yds 

30  dollars  buy  ? 

20  :  so  :  :  6 

S  yds.     g 

6 

9  :  6  :  :  so 
6 

2,0)18,0 

9)180 

Ms.  »9 

Ans.  myds. 

4.  If  Scwt.  ofsu»ar 

cost  Si.  8s.  w 

hat  will  11  cwt.  1  qr. 

§4  lb.  cost  ? 

S  cwt.  SI.  85.  C.  qr. 

lb.            lb. 

s. 

112          20        11  1 

24    As   336p  :  168  :  :  1284^^, 

-^        4 

168 

S^6Z&.   168s,     — 

\ 

4^ 

10272 

28 

r^' 

7704 

...^ 

/ 

1284 

564 

—  ■(2,0) 

92 

536)215712(64,2 

2016     

1284  » 

32Z.2S, 

1411  Ms. 
1344 

672 

672 

ft* 

^ 

* 

;\  n  cac  ;;.aii:  ^>lv  sc^^fLiUngs  ccfi^  4s.  Ga.  wluii  will  19 

6,-1.  "    '        I  pair  oT  &  ■  -vtZ.  6s.  what  will  one 

piiir  Civ  ^ns.  4s.  6(f. 

r.  Ai  iOk..  per  pound,  what  is  the  value  of  a  firkin  of 
t>utter,  weight  56  pounds  ?  ^?is.  £^  95. 

8.  Kuw  iiiuch  sugar  caiuyou  buy  for  23Z.  Ss.  at98.  a 
pound  ?  Jins.  5C.  ^rs. 

9.  Bought  S  chests  of  sugsir,  each  9  cwt.  2  qrs.  what 
do  they  come  to  at  2^.  5s.  j^ercwt.  ^  Ans.  £17i. 

10.  if  a  man's  wages  arc  751.  10s.  a  year,  what  is  that 
a  calendar  month  ?  Ji'iis,  £6  5s,  lOd. 

1 1.  If  4^  tons  of  hay  will  keep  5  cattle  over  the  v/inter : 
Iiow  many  tons  will  it  take  to  keep  25  cattle  the  same 
time  ?      "  ^Ins.  37^  tons. 

12.  If  a  man's  yearly  income  he  20SL  Is.  wl^.at  is  that 
a  day  ."  Jlus.  lis.  4d.  S-^/^qrs, 

IS.  If  a  man  sp^inds  Ss.  4d.  per  day,  how  muchi?  that 
a  year?  t,^ns.  £G0  ICs.  Sd. 

14.  ilCidlngat  12s.  Cd.  per  week,  how  long  will  o£L 
IB.?,  last  me  .^  .•^?2.s.'  1  ?/^/?r, 

15.  A  owes  B  C>475l.  Dut  I>  compounds  with  him  iht 
73s.  4d.  on  the  pound  ;  pray  what  must  he  receive  for 
his  debt  r  ^       Ms.  £2S\6  ISs.  4d. 

16.  A  j^klsmith  sohl  a  tankard  for  SL  12s.  at  5s.  44* 
gpr  r?r;i:c:;,  what  was  the  weight  of  the  tankard  ? 

»  ^^ns.  9.11).  ?>oz.  5pivh 

17.  i;2cvf.  S  qrs.  21  Io\  of  Sugar  cost  6L  Is.  8d.  w^hat 
cDst  35:^  cut  r  ,lns.  £7o. 

18.  Bought  10  piecei^  of  cl;)thj  each  pj ecc  cnntainin^  . 
9^  }Tu-ds^  at  Its.  4i  ponce  jVCr yard  5  what  did  thc.M'hole, 


FIH)ERAL  MONEY. 

KOTE  1.  You  must  state  the  question,  as  taught  in 
the  Rules  foregoing,  and  after  reducing  the  first  and  third 
fi!rms  to  the  same  name,  &:c.  you  may  multiply  and  di- 
vide according  to  ^i\^  rales  in  decimals  ;  or  by  tl:e  rule^ 
^or  multiplying  and  divMiug  Federal  MojiQ".  \ 


..(]:- 


:s3 


19,  If  r  yds,  of  cloth  cost  15  dollars  47  cents,  what 
willl3yds/cost?  Vds.    Scis.     yds, 

7  :  15,47  :  :  12 
12 


7)185,S4 


•3m5,  26,53=8-<3.  52n\^. 
•But  any  sum  in  dollars  and  cents  may  be  written  down 
CS  a  whole  number,  and  expressed  in  its  lowest  denomi- 
nation, as  in  the  following  example :     {See  Reditctipn  of 
Federal  Moiiey^  page  67.) 

£0.  What  will  1  qr.  9  lb.  sugar  come  to.  at  6  dollars 
45  Cts.  per  cwt.  ? 

gr.  Ih,  Ih,       cts,         lb. 

19  M  112  :  645  :  :  37 


as 


r.r 


lb.  4515 

1935 


-^ cts. 

112)23865(213+  .-3ws.=g2,  IS. 
224 


345 


\ 


9 

NOTE  2.  When  ihe  first  and  third  numbers  are  fede- 
ral money,  you  may  annex  cyphers,  (if  necessary)  un^l 
j!t)U  malie  tfieir  decimal  plrxcs  or  figures  at  the  right 
hand  of  the  separatrix,  equal:  which  will  reduce  them  to 
a  Hke  denonr?inatioa.  Then  you  may  multiply  and  di- 
vide, as  in  whole  numbers,  and  the  quotient  will  expreis 
the  answer  in  the  least  denomination  mentioned  in  _tli« 
f^cmd,  or  mi^id'e  ter)^. 


IQi 


nULE   of  THUEE    PIRECT. 
EXAMPLES. 


SI.  If  3  dolidfs  will  buy  T  yards  of  cloth,  how  many 
Jjfcrds  can  I  buy  for  ISO  dollars,  75  cents  ? 
cts.     ijds,       cts. 

As  SO0  s  7  f :  nor 5 
7 

yds. 

500)84525(2811  *^m: 
22.  If  1  ■^.  lb.  of  Tea  cost  6  dols.  600 

rS  cts.  and  9  mills,  what  ^yill  5  lb.  

cost  at  the  same  rata  }                          2452 
ib.     mills.         Uh                        2400 
As  12  :  6739  :  :  5  


525 
500 


12)35945 


Zct^'in.  225 


^Ins   2828-F-mi/I^j=2582,S.  4 

900(a2i*5* 
900 
S    cfs.  — 

23.  ir  a  ma/i  lays  out  ]2I.  25  in  merchandize,  and 
thereby  gains  39  dolhirs,  51  cts.  iiovv  much  will  he  gain 
by  laying  out  12  dollars  at  the  same  rate  t 

Cents,      cents.      cenis.  v 

As  12123  :  3951  :  :  1200 
1200 

cts,    S  cts. 

121£3)4741200(391  =55,91  .in$. 
36369 


110450 
109107 

15250 
1S123 

nor 


^ 

wi 


•latTIK  0?  THREE  ©IREOT-  105 

54.  If  the  wages  of  15  weeks  come  to  64  do!-..  lO  rts. 
©at  is  a  year's  wages  at  that  rate  ? 

Jlns.  8222,  5.:......  ..... 

55.  A  man  bought  sheep  at  1  dol.  11  cts.  per  head,  to 
the  amount  of  51  dols.  6  cts.  j  liow  many  sheep  did  he 
buy  ?  ,  .4«s.  46. 

26.  Bought  4  pieces  of  cloth,  each  piece  containing  31 
yards,  at  l6s.  6d.  per  yard,  (^New-England  uxnency) 
what  does  the  whole  amount  to  in  federal  money  ? 

Ans.  8341. 
Sr.  When  a  tun  of  wine  cost  140  dollars,  what  cost  a 
quart  ?  dns.  IScts,  S^-^m. 

23.  A  merchant  agreed  with  his  debtor,  that  if  he 
would  pay  liim  down  65  cents  on  a  dollar,  he  would  give 
him  up  a  note  of  hand  of  249  dollars,  88  cts.  I  demand 
Ti'hat  the  debtor  must  pay  for  his  note  ? 

Ans.  8162,  42ci<J.  2m. 
29.  If  12  horses  eat  up  SO  bushels  of  oats  in  a  week, 
tow  many  bushels  will  serve  45  horses  the  same  time  ? 

Ans»  112^  bushels. 

50.  Bouglit  a  piece  of  cloth  for  848  27  cts.  at  1  dollar 
19  cents  per  yard ;  how  many  yards  did  it  contain  ? 

Ans.  40yds.  2qrs.  y\^^. 

51.  Bought  3  hhds.of  sugar,  each  weighing  8  cwt.  1  qr. 
IS  lb,  at  7  dollars,  26  cents  per  cwt.  what  come  they  to  ? 

Ms.  8182  let.  8m. 
^  ^.  What  IS  the  price  of  4  pieces  of  cloth,  tlie  first 
piece  containing  21,  the  second  23,  the  third  24,  and  tlie 
liwrth  sr  yards  at  1  dollar  43  cents  a  yard  f 

Ms.  8135  '66cfs.  21+25+244-2r=0%!/>-. 
^.  Bought  Shhds.  of  brandy,  contaiiiing  6}..  'H,  f^:Li 
gallons,  at  1  dollar,  38  cents  per  gallon,  I  demand  .'.ow 
much  tiiey  amount  to  ?  Ms.  S255,  Vj/is. 

34.  Suppose  a  gentleman's  income  is  1836  dollars  a 
year,  and  lie  spends  3  dollars  49  cents  a  day,  one  dav 
with  another,  now  much  will  he  have  saved  at  the  yea^-'s 
end  ?  Ms.  8562,  ]  5cU. 

35.  If  m^  horse  stands  me  in  20  cents  per  day  kcep- 
tng,  v/hat  Avill  be  the  churgc  of  11  horses  for  the  yeai-,  at 
Inatratef?  ^  Ms.  g803. 


So.  A  merchant  boijiglit  14  pipes  of  wine,  and  Is  allo^vH 
ed  6  months  credit,  but  for  ready  money  gets  it  8  cents  a 
gallon  cheaper ;  how  much  did  lie  save  by  payirsg  ready 


EX^MPLES^i 


Cr.    Sf^L: 


at 


was  my  pi^ri  ^;i  t.io  niouey  r  .iVh.  £-^Ji 

33.  if  -^5-  of  a  vshlp  cost  TSl  dolhirs  25  cents,  M'hat  is 

t^ie  whole  wortli  ?  S 

As  5  :  r8l.*:^5  :  :   i6  :  2500  .2;zi?. 
59.  If  I  buy  54  yards  of  cluih  fi-^T  S\L  10s.  what  did 

'^,  cost  per  Eli  Englisli  ?  dn<i,  14s,  7d. 

40.  Bought  of  Mr.  Grocer,  11  cwL  3  qrs.  of  sugar,  at  8 
dollars  12  cents  per  cwt.  and  gave  idiii  James  Payweirs 
note  for  19^.  Ts.  (Ncw-Englau<l  currency)  the  rest  I  pay 
in  cash;  tell  nie  how  inaiiy  uidiars  vvill  make  up  W\q 
balance  t  ^  .Qns,  S50,  91  cis. 

41.  If  a  stair  5  feet  long  casts  a  shade  on  level  groumi 
8  feet,  wluit  is  the  lieight  of  that  steeple  whose  shade  at 
the  same  time  measures  1 81  feet  .^  Ans.  llS}ft, 

42.  If  a  gentleman  has  an  income  of  300  Englisii  guin- 
eas a  year, how  mrich  may  he  spend,  one  day  with  anoth- 
er, to  lay  up  500  dollars  at  tiie  yearns  end  ? 

dns.  S2,  4Gch,  5m.    .^ 

43.  Bought  50  pieces  of  kerseys,  each  34  Klis-Flemish,  *' 
at  8s.  4d.per  Ell-Ei^lish;  wliat  da)  tiic  whole  coi;t.^ 

Jhis.  £4^25. 

44.  Bought  200  yaitls  of  canitjiick  for  00/,  but  being 
damaged,  i  am  willing  to  lose  71,  10s.  by  tl\e  sale  of  it ; 
what  must  I  demand  per  Ell -English  ?    Ans,  lOs,  Sfr/. 

4.5.  Hov/  many  jneces  oi  ii<.liaad,  each  20  Ellg-FIem- 
ish,  may  I  have  fxM*  25/.  8s,  at  6s.  6d.  per  Ell-English  ? 

An^,  6  lueces, 
46.  A  merchant  bought  a  bale  of  ch^th  containing  £-10 
yards,  at  the  rate  of  7^  dollars  for  5    yards,  and  soiii  it 
again  at  the  rate  of  1 H  doHajs  for  7  yard^jj  did  he  gaia 
or  loae  by  the  bargain,  and  how  much  ? 

.fi:t$,  lie  gained  S25,  71  .^/.«:.  Ad:.  -^ 


RULE   OF    THREE  DiREfJT.  1^? 

47.  Bougjit  a  pipe  of  wine  for  84  dollars,  and  fonad  it 
had  leaked  out  12  gallons  ;  I  sold  the  remainder  at  12 J 
cents  a  pint  5  what  cfid  1  gain  or  lose  ? 

Ms,  I  gained  S'30. 

48.  A  gentleman  bought  18  pij}Gs  of  wine  at  12s.  6d. 
(New -Jersey  c\irrencj)  per  gallon  5  how  many  dollars 
will  pay  the  purchase  ?  •^ns.  g3780. 

49.  Bought  a  quantity  of  plate,  weighing  15  lb.  11  oz. 
ISpwt.  17  gr.  how  many  dollars  will  pay  foritj  at  the 
rale  of  12s.  7d.  New -York  currency,  per  ounce  ? 

Jins,  |g;301,  5Uct^  2j%w. 

50.  A  factor  bought  a  certain  qiiautity  of  br(jadc*oth 
and  dru^et,  which  together  cost  81/.  the  quantity  ot 
broadcloth  was  50  yards,  at  1 8s.  per  yard,  and  for  every  5 
yards  of  broadcloth  1^  iiad  9  } ards  of  chugget ;  I  demand 
how  many  yard«  of  drugget  he  iiad,  and  wha^  it  cost  him 
per  yai-d  r  3ns.  90  yards  at  Ss,  per  yard, 

51.  If  I  give  1  etigl  65*2  dollars  8  dimes,  2  cents  and  5 
tnills,  for  675  tops,  now  many  tops  will  19  mills  buy  ? 

•fins,  I  top 
53w  Whereas  an  eagle  and  a  cent  jr.st  tliree  score  yards 
did  buy, 
H6w  many  yards  of  that  same  cloth  for  15  cliin^.3  had  I  P 

•^ns.  Si/ds,  Sqrs,  Sna.+ 
55.  If  the  Legislature  of  a  State  grant  v  ta?:  of  8  milU 
Qfi  the  dollar,  how  much  must  that  man  pa  -  •  9  d  0!  - 

lars,  75  cents  on  the  list? 

dns.  S2,  55cts,  8r»i, 

54.  If  100  dollars  gain  6  <lollars  interest  in  a  yeai, 
how  much  will  49  dollars  gahi  la  the  same  time  .^ 

55,  If  60  gallons  of  watftr,  in  one  hotti-.  6^1  iuu:  &  c;^ 
tern  containing  SOOgallofis,  ano  by  a  »>ipe  in  t-m  eMmi^ 
S5  gallons  run  out  in  an  liour  :  in  what  tiiae   will  n  b?. 
fdl»*d  ?  ^ns,  in  l^houTH, 

ol),  A  and  B  depart  Iron  t{>e  sanie  piace  ana  trard 
the  game  wad;  but  A  ipoji  5  (ir'r?  befai-e  I  J.  at  ll-e  mh- 
of  15  miles  a  day  ;  1^  follows  vit  trie  »"itfi  cf  siO  mils*  ^ 
day  'f  what  dis*tanr.e  must,  he  travel  to  ov«M'take  A  r 

Jns.  500  mlUB* 


11 


KULE  OF   THREE  iiJVKRSE. 

JLE  OF   THREE   INVERSE., 

^  ,ip  iff  Three  Inverse,  teaches  by  liavJng  ihxi^ 
r  •*-'■''  4  ''Jiirth,  which  shall  have  the  same 
.    u?i>  as  the  iirsthas  to  the  third. 
)  >re   equires  more,  ov  less  requires  less,  the  quesT* 
iio    '  -'iHigs  to  the  Rule  of  Three  Direct : 

fiut  it  more  requires  less  or  less  requires  morcy  the 
question  belongs  to  the  Rule  of  Three  Inverse  ;  \thicli 
may  always  be  known  from  the  nature  and  tenor  of  tlie 
question.  '  For  Example  ; 

If  2  men  can  niowa  field  in  4  days,  how  many  days 
will  It  require  4  men  to  mow  it  ? 

771  en  days  meyt 

1.  If  2  require  4  how  much  time  will  4  re* 
quire  ?  Anstver,  £  days.  Here  more  requires  Ifis?,  Vii^ 
the  more  men  the  less  time  is  xi^qyii'e^ 

men  days  ^ii^tt 

2.  If  4  require  2  how  much  time  ^vill  £i  f^* 
quire  ?  Answer;,  4  days.  Here  less  requires  more,  vi«. 
the  less  the  number  of  men  are,  the  more  days  are  requir- 
ed— therefore  the  question  belongs  to  Inverse  Proportion* 

RULE.  ^ 

1.  State  and  reduce  lint  terms  as  in  the  Rule  ofThrefc 
Direct. 

2.  Slultiply  the  first  and  second  terms  together,  and 
divide  the  product  by  the  third ;  the  quotient  will  be  thJfe 
answer  in  th^  same  denomination  as  the  middle  term 
was  reduced  into. 

EXAMPLES. 

1.  If  V2  m£n  can  build  a  wall  in  20  day^,  How  iriaj^y 
men  ra:i  do  the  same  in  8  days  ?  *flns.  SO  ttien, 

2.  If  a  manpcsforms  a  jounic  j  in  5  days,  when  iFie 
tlav  1  UUioia's  long,  in  how  many  days  v;iil  he  perform 
it  {vLen  the  dav  is  In- 1  10  hours  long  ?       dns.G  iUijiB. 

">    What  h^nj^  :  :  vd  7i  iilcHes  tvido*  will  maice  Si 


raACTXOE.  l(i^ 

4.  If  live  dollars  will  paj  for  tlic  carriage  of  2  cwt.  150 
miles,  how  far  may  15  cv;t.  be  carried  for  the  same  mo- 
ney ?  •''his,  20  miles, 

5.  If  when  wheat  is  7s,  6d.  the  bushel,  the  penny  loat 
will  weigh  9oz.  what  ought  it  to  weigh  when  wheat  is  6s. 
pep  bushel  ?  Ms,  Uoz,  5pivt. 

6.  If  30  hushels  of  grain,  at  50  cts.  per  bushel,  will 
pay  a  debt,  how  maiw  bushels  at  75  cents  per  bushel,  will 
pay  the  same  ?         "  •^na,  20  bushels.  ^ 

7.  If  lOOl,  in  12  months  gain  61,  interest,  what  princi- 
pal will  gain  the  same  in  8  months  ?  Jins.  £150. 

8.  If  11  men  can  build  a  house  in  5  months,  by  work- 
ing 12  tours  per  day — in  what  time  will  the  bame  num- 
ber of  men  do  it,  when  they  work  only  8  liours  per  day  ? 

Jiizs,  7h  intriUihS, 

9.  Wkai  nuiuber  of  men  must  be  employed  to  fiidsh  in 
5  days,  what  15  men  would  be  £0  days  about  ? 

w2?i5.  60  mtn. 

10.  Suppose  650  men  are  in  a  garrison,  and  their  pro- 
visions calculated  to  last  but  two  months  ;  how  many  men 
must  lea\  e  the  garrison  tliat  the  same  provisioiis  may  be 
gufficient  fur  those  who  renuiin  live  months  ? 

Jlns.  ^90  men, 

11.  A  regiment  of  soldiers  consisting  of  850  men  are 
to  be  clothed,  each  suit  to  contain  3^  yds.  of  cloth,  which 
is  li  yards  Wide,  and  lined  witli  shalloun  |  yard  wide^ 
liow  many  yards  of  sliaiicon  will  comolete  ^wo.  liriing? 

Jim,  e94iyds,  ^qrs,  ^na.  ' 


FKACTICE.  ^-s 

Practice  is  a  contraction  of  t:ie  Rule  of  'ilirce 
I'iirect,  v/hen  the  first  term  happens  toi.e  an  unit  or  owe, 
and  is  a  concise  method  of  resolving  ir;ost  questions  that 
occur  in  trade  or  business  where  monev  is  reckoned  in 
pounds,  shillings  and  jXiuce ;  but  reck^'jji'ng  in  federal 
Money  will  render  this  rule  almost  v.sr.lis's  :  for  vhirl; 
reason  I  shall  net  enlar^^e  so  much  o-.  Oir  ;in])ject  as  ma-r 
Dv  ether  wiiters  We  done. 


no 


WiACTICE. 


Parts  of  a  Shiihne;, 
is       !j 


Tables  vf  Jiliquot,  or  Even  Farts. 


6 
4 

S 

2 


1 
1 


Parts  of  2  Shillings. 
Is.     is     i 


8d. 
6d. 
4d. 
3d. 
2d. 


Parts  of  a  PouTid.l   Parts  of  a  cwt. 


d, 
10  0 
6  8 
5  0 
4  0 
3  4 
2  G 
1  8 


£•    1 

i&. 

cwU 

i 

56 

ii     h 

^     1 

28 

=  i 

i 

16 

4 

i 

14 

i 

1 

ir 

1 

•B 

i 

15- 

I 

¥ 

tV 

The  aliquot  part  of  any  number, 
is  sucli  a  pai't  of  it,  as  being  taken  a 
certain  number  of  times,  exactly 
makes  that  number. 

CASE   I. 

When  the  price  of  one  yard,  pound,  &c.  is  an  even  part 
of  one  shilling. — Find  the  value  of  the  given  quantity  at 
Is.  a  yard,  pound.  Sec.  and  divide  it  by  that  even  part 
and  the  quotient  will  be  the  answer  in  shillings,  &c. 

Or  find  the  value  of  the  given  quartity  at  2s.  per  yard, 
&c.  and  divide  said  value  by  the  even  part  whicli  the 
given  price  is  of  2s.  and  the  quotient  will  be  the  answer 
in  shillings,  &:c.  whicli  reduce  to  pounds. 

N.  B.  To  find  the  value  of  any  quantity  at  2s.  you  need 
only  double^ the  unit  figure  for  shillings;  the  other  fig- 
ures will  be  pounds. 

EXAMPIUilS. 

1.  "What  will  46U  yards  of  tape  come  to,  at  1  id  per  yd,  r* 
s.  d, 
Ud.  (  ^  I  461  6  value  of  461  6  yds.  at  Is.  per  yd. 


5,7  V>i 


£2  ITS,  8icZ.  value  at  l^id. 
What  cost  25Glb.  of  cheese  at  8<i.  per 


noujid  ? 


Sd. 


j  /:2j  12s.  value  of  2561b.  at  2s.  per  lb. 


xC^^ 


8f?.  va!u<it  of  8d.  per  pound 


PRAOTICE. 

Yarck,  per  yar'dy 

486i  at  Id.  Answer}$. 

862  at  2il. 

911  at  3d. 

749  at  4d 

113  at  6d  2  16 

899  at  8d.  29  19    4 

CASE  \L 

When  the  price  is  an  even  part  of  a  pound — Find  the 
value  of  the  given  quantity  at  one  pound  per  yard,  &c. 
and  divide  it  by  that  even  part,  and  the  quotient  will  be 
the  answer  ki  pounds. 

EXAMPLES. 

WTiat  will  129^  yards  cost  at  2s.  6d.  per  yard  ? 
s,d.  £.  s,  £. 

2  6  I  I  I  129  10  value  at  1  per  yard. 

dns,  £  16  3s.  9i.  value  at  2s.  6d.  pw  yard. 

Yds.        #.   rf.  r.  s.  d. 

123     at  10  0  per  yard  Jimwerf     61  10  0  ^ 

687^  at    5  0      —  171  17  6  W 

21U  at     4  0      ■—  42     5  0 

543     at     6  8      —  181     0  0 

127    at    3  4      —    .  21     3  4 

461     at     1  8      —.  38     8  4 

Note.  Wiien  the  price  is  pountls  only,  the  given  quan- 
tity multiplied  thereby,  will  be  the  answer. 

Example. — U  tons  of  hay  at  Al.  per  ton.     Thus     li 

4 

Ans*  £44 

CASE  III. 

When  tlie  given  price  is  any  number  of  shillings  ua 
der;  20. 

1.  When  the  shillings  are  an  even  number,  nnlltiply 


no 


IVIiACTlCE. 


Parts  of  a  Shiilmg. 
d.  s. 

is       !j 


Tablet,  *iff  Aliquot^  or  Even  Farls, 


6 
4 

S 

2 

u 


1 

"iS" 


Parts  of  2  Shillings. 
Is.     is 
8d.    = 
6(1. 


4d. 
3d. 
2d, 


\ 

1 


Parts  of  a  Pouiid.l   Parts  of  a  cwt. 


5.  d, 

10  0 

6  8 

5  0 

4  0 

3  4 

2  G 

1  8 


£• 
i 


Ih, 

5(S 


liB 


* 

%» 

i 

16 

1 

"5 

14 

^' 

r 

1 

IT 

tV 

cwU 


The  aliquot  part  of  any  number^ 
is  such  a  part  of  it,  as  being  taken  a 
certain   number   of  times,   exactly 
^2^        makes  that  number. 

CASE   I. 

When  the  price  of  one  yard,  pound,  &c.  is  an  even  part 
of  one  shilling. — Find  the  value  of  the  given  quantity  at 
Is.  a  yard,  poiinJ,  Sec.  and  divide  it  by  that  even  part 
and  the  quotient  will  be  the  answer  in  shillings,  &c. 

Or  find  the  value  of  the  given  quartity  at2s.  per  yard, 
&c.  and  divide  said  value  by  the  even  part  whicli  the 
f^iven  price  is  of  2s.  and  the  quotient  will  be  the  answer 
in  shillings,  &c.  which  reduce  to  pounds. 

N.  B.  To  find  the  value  of  any  quantity  at  2s.  you  need 
only  double^ the  unit  figure  for  shillings;  the  other  fig- 
ures will  be  pounds. 

EXAMPUiS. 

1.  What  will  46U  yards  of  tape  come  to,  at  1  id  per  \s^.  .^ 
s.  d.'^     , 
lid.  [  ^  I  461  6  w'u"',}:.^  of  461  h  yds.  at  Is.  per  yd. 

5,7  ?>\ 


£2  I7s,  8 i^.  value  at  Ud. 
,  What  cost  S56lb.  of  cheese  at  8d.  per  pou^id  ? 
Sd.  [  ^^  I  /:2j  12s.  value  of  2561b.  at2s.  per  lb. 

f^2  4  of  8d.  per  pound 


PRAOTICE. 


r' 


Yards, fZTyar'dy  £*  s,  d.  ^ 

486i  at  Id.  Jinswer^.    2    0  6J 

862    at  'M.  '758 

911     at  3d.  11     7  9^^ 

749    at  4d  12    9  Wff 

113     at  6d  2  16  6 

S99    at  8d.  29  19  4 

CASE  11. 

When  the  price  is  an  even  part  of  a  pound — Find  the 
value  of  the  given  quantity  at  one  pound  per  yard,  &c. 
and  divide  it  bj  tfiat  even  part,  and  the  quotient  will  be 
the  answer  ki  pounds. 

EXAMPLES. 

WTiatwiii  129i  yards  cost  at  2s.  6d.  per  yard  ? 
s.d,  £.  s,  £. 

2  6  I  j  I  129  10  value  at  1  per  yard. 

^ns,  £16  3s.  9^^.  value  at  2s.  6d.  pwyard. 

Yds.        #.   rf.  r.   s.  d. 

123     at  10  0  per  yard  Jlmiverf     61  10  0  4l 

687^  at    5  0      —  171  17  6  v*' 

211i  at     4  0      —  42     5  0 

543     at     6  8      ■—  181     0  0 

127    at    3  4      —    >  21     3  4 

461     at     1  8      --.  38     8  4 

NoTE.^  Wlien  the  price  is  pounds  only,  the  given  quan- 
tity multiplied  thereby,  will  be  the  answer. 

Example. — U  tons  of  hav  at  4L  per  ton.     Thus     li 

4 

Ms.  £44 

CASE  III. 

When  tlie  given  price  is  any  number  of  shillings  un 
dei;  20. 

1.  Whe.n  the  shillings  arc  an  even  number.  muUipIy 


$ 


vn  A  or  I  OF., 


<Tie  qu|nf;ltjl>y]:airi]ie  nm-ibrvf^F  ^Mn^:-gs.  and  double 
the  firsi  ilgure  of  the  product  '  \^\s ;  and  the  rest 

of  the  product  will  be  pounds. 

2.  "^^c  shillings  be  odd;  niulliply  the  <]uaTititj  by  Iho 
wiioleQpimber  of  .shillings,  and  the' product  wilf  be  tlie 
answer  in  shillings,  which  reduce  to  poiimL-. 


1st.  124  Yds.  at  8p. 


£49  12.-?.  Ms. 

2,0)92.4 

Yds.                           £.     f. 

562  at    4s.     Jus.  112    & 
378  at    2s.                57  IG 
915  at  14s.              059    2 

rAM  Ms. 

fcU.     ' 
^,7^2  at  lis. 
264  at     9s. 
2oO  at  I6s. 

SiwT.,  5^4  12 
200  00 

CASl 

3  IV. 

Wlien  tiic  given  price  is  pence,  o?-  pence  andTarthin^s, 
and  not  an  even  part  of  a  shilling — Find  tne  value  of  flie 
given  quantity  as  Is.  per  yard,  &c.  which  divide  by  the 
greatest  even  part  of  a  snilling  contained  in  the  given 
price,  and  take  parts  of  i\\^  quotieni  for  the  reriiaincier  of 
tlie  price,  and  the  siun  of  these  several  quotients  vrili  be 
tTie  answer  in  shillings.  &c.  which  reduce  to  pomnds. 

KXAMPI/ES. 

What  will  245  lb.  of   raisins codk  i>  .  u^  .  ^v;.  ,.vrlb.  ? 
s.      d. 


6d. 
Sd. 


^ 


15     0  value  of  245  lb.  at  Is.  per  pound. 


122  G  valweof  do.  at  (kl.  per  !b. 
Gl  S  valucof  do.  at  5d,  perlb. 
i  5     5  J  value  of  do.  at  i'd.  ];or  lb. 


2,a)19,9    OJ 


M^,  £9  19  0;'  value  cf  tlic  v.  h(j!e  at  9id.  per  lb 


rilACTICE. 

Sr£  at  1}  .in^.    '^  14     3 

325  at  2i    .<         S     0  1 U 

"^^7  at  42    '       15  10     U 

576  at    7i  Ms 
541  at    9^ 
672  at  Hi 

11 

1300 

32  18  0 

C\SK  V. 

Wken  the  price  is  fyhilliiigs,  pence  and  farthings,  and 
not  the  aliquot  part  of  a  pound — Multiply  tiie  given  quan- 
tity by  the  shillings,  and  take  parts  for  the  pence  and  far- 
mings, as  in  the  foregoing-  cases,  and  add  them  togetlier; 
tlie  sum  will  be  the  answer  in  shillings. 

EXAMPLES. 

1.  What  AviU  246  yds.  of  velvet  come  to,  at  7s.  Sd.per 
yard?  s,   \L 

Sd.  [  d  {246  0  value  of  246.  yards  at  1«.  per  yd. 


1722  0  value  of  do.  at  7s.  per  yard. 
61  6  value  of  do.  at  Sd.per  yard. 


2,0)178,  3  6 


^iis,  £89  3  6  value  of  do.  at  7s.  per  yard. 

ANSVVKRS. 

5.  cL  £.    s.   d. 

2   What  cost  159  yds.  at    9  10  per  yd.  }  68     6  10 

S.  What  cost  146  yds.  at  14     9  per  yd.  ^  107  13     6 

4.  Wliat  cost  120  cwt.  at  1 1     3  per  cwt.  }  (S7  10 

5.  What  cost  127  yds.  at    9     8^  per  yd.  ^  61  12  11  ^ 

6.  What  c«st  49^  ib«    at    3  lU  perU.  ?  9  15  lU 

CASE  VI. 

When  th**  price  aiid  qur^ntity  given  are  of  several  de- 
noiiiinations-^^Multiply  the  price  by  the  integers  in  the 
gtven  quantity,  and  take  part?  for  mt  rest  from  the  price 
of  an  integer ;  whicli  added  together  will  be  ihe  '^«*r«' 
This  is  applirnble  ^c  FedcraJ  ^lonev 


n4 


TAKE    AND    TRV/IT. 


i:XAMPL3<:S. 


1.  What  cost  5cwt.  Sqrs. 
I4lb.  of  raisins,  at  2L  lis. 
fid.  per  cwt.  ? 

-'     ~  \£,      s.      d. 

Sqrs     ^2      11      S 

i  5 


iqr. 
14  1b. 


2.  What  cost  9cwt.  Iqr. 
8lb.  of  sugar,  at  8  dollars, 
65  cts.  per  cvvt.  ? 

S  cts. 
1  qr.     -i      8,65 
9 


7  ]]>. 
1  lb. 


?.,16^5 
,5405 


Ms.  £15       5  '    6,v|  Ms,  g80,6S'03 

€.  grs.   lb.  answers. 

7  5     16  at  S9,  58cts-.-^>ei-e'^rt.-"-.-  ^75,  6lcts.  Sm. 

5  10  at  2/.  irs.  percwt.       ;         /•14  19s.  Sd. 

14  S      7  atOZ.  ISs.  8d.  pcrcvrt.         £\Q  2^,  5R 

32  0       r  at  S6,  r>4cts.  per  cwt         ^76.  4rcif5.  6:?u 

0  0     24  at  Sll,  91cts.  per  p\vt.     S2,  55cis.  2-j^m. 


TARE    AND    TRKTT. 

,1  ARE  and  Trc it  are  practical  Rules  for  dcductinj; 
certain  allowances  ^v]lich  are  made  bj  niercbants.  in 
))uyingand  selling  goods,  Sec.  by  weight :  in  VvliicL  are 
noticed  the  following  particulars  : 

1.  Gross  JVeightj  wiiich  is  the  whole  weight  Oi  any 
sort  cf  goods,  together  with  the  box,  cask,  or  bag,  &c. 
which  contains  them.  ^99^ 

2.  Tarc^  whicli  is  an  allowance  made  to  tT^e  buyer 
for  the  weight  of  the  box,  cask,  or  bag,  &c.  whicli  can- 
tains  the  goads  bmightjand  is  cither  at  soniuch  per  box 
&c — or  at  so  much  per  cwt.  crat  so  much  in  the  whole 
gross  weight. 

S.  Trett,  whicli  is  au  allowance  cf  4  lb.  on  every  104  lb 


•3  AKK    ANI»     TKKl  I.  >  13 

4.  CTofl^,  v/liicli  is  rh  allowance  made  of  ;I  ib.  upon. 
GV4?ry  3  cwc. 

5.  Suttte,  is  ^vhat  remains  after  one  or  tv/o  allowances 
have  been  deducted. 

CASE  I. 

When  the  question  is  an  Invoice. — Add  the  gross 
>veig;hts  into  one  sum  and  the  tares  into  another  :  then 
subtractthc  total  tare  from  the  whole  gross,  and  the  re- 
liiaiiider  will  he  tlie  neat  weight. 

KXAMPLKS. 

1 .  V/hat  is  tlie  neat  vreiglit  of  4  hcgslicads  of  Tobacco 

marked  with  the  gross  \^'eight  as  follov/s  : 

C.     cp\     lb,  Ih, 

No.  l-~9      0       13    Tare    100 

o_^      3         4      —        95 

S  -^  7       X         0       ^        83 

4--G       3       £5       -—        81 


Vv'hole  gross  52      0       13  559  total  tare. 

Tai-e  359  ]b.=5       0      23 


Ms.  £8      3  18  Kcat. 
2.  What  is  tlie  neat  weight  of  4  barrels  of  Indigo,  No. 
and  weight  as  foliov/s : 

C.  qr.  lb,  Ih 

^^o.  1  —  4     1     10  Tare    56"^ 

i2  —  5     5     iVZ  —      £9  ( 

5  — .  4     0     19  —      52  f          cwL  qr.  U 

4  —  400  —      S5J  Ms.  15    0    11 

CASE  II. 

When  the  tare  is  at  so  much  per  box,  caskj  bag,  &c.— 
Multiply  the  tare  cf  1  by  the  number  of  bags,  bales,  &;c. 
the  product  is  the  v/hele  tare,  which  subtract  from  the 
gvoas,  and  the  remainder  will  be  the  neat  weight. 

KXAMrLHa. 

1.  In  4  hhfls.  of  sugar,  each  weighing  lOcwt.  l({r.  15lb* 
gmas ;  ti:Lr2  T5\b.  per  iihd.  ho\7  much  rscat  ? 


J  T5  1AK14    AND    TRETT. 

^Toss  weight  of  one  hhd. 


10 

qrs. 
I 

15 

4 

41 

75X4=2 

2 

4 
£0 

4  gross  weight  of  the  whole. 
£0  whole  tire. 

\flns.  58      3      12  neat. 
2.  What  is  the  neat  v.eight  of  7  tierces  of  rice,  each 
weighing  4cwt.  Iqr-  9Ib.  ^^ross,  tare  T>er  iTerce  34lb.  ? 

.iiw,^9.Sa  Oqr.  2llh. 
5.  In  9  firkins  of  butter,  each  weighing  2([rs.   12lb. 
gross,  tare  lilb.  per  firkin  ;  how  much  neat  ? 

.^ns,  40.  Qqrs,  9lb, 

4.  In  241  bis.  of  ligSj  each  Sqrg.  191b.  gnrss,  tare  lOlb 
per  barrel;  how  many  p(*untls  neat  .^         *'^Ins.  22413. 

5.  In  16  bags  of  pepper,  each  85lb.  4oz>  gross,  tare  per 
bag,  Sib.  5o7>, ;  hov/  manyponnds  neat.^      •Bus,  1311. 

6.  In  75  l)arreis  of  figs,  eacli  2([is.  52rlb.  gross,  tare  in 
the  whole,  507\h. ;  how  much  neat  weigh.t  ? 

.ins,  50  C.  \qv. 

7.  What  is  the  neat  v/eidit  of  15  hlwls.  of  Tobacco, 
each  weighini*;  Tcwt.  \p^v.  ISlb,  tare  lOOib.  per  hhd.  ? 

Am,  or  a  Oqr,   \)lh, 

CASE  ilf. 

When  the  tare  is  at  so  much  per  cwt. — Divide  the 
gi'033  weight  by  Hii^  aliquot  part  of  a  cwt.  for  the  tare, 
which  subtract' from  the  gross  and  tha  remainder  will  be 
neat  weight. 

ii::A:.i?Lr.s. 

1,  What  is  the  ne.it  weight  of  44cwt.  Sqrs.  iSlb 
gross,  tare  14Ib.  ptr  cv/t.  ? 

C.     //r.>.  lb. 

!  14lb.  [  }  I  44    '3  IG    gross. 

-5      2  122  tare. 

^jHf.  Sf9      1        Sk  neat. 


TAiiv    .         ...  .  ;;■ 

;:.  What  is  the  neat  'weight  of  9  hhds.  of  tobacco,  each 
weighing  gross  8cwt.  Gqrs.  141b.  tare  I6lb.  per  cwt.  ? 

./?r?s.  eScivt.  U;r.  24/6. 
5.  What  is  the  neat  weight  of  7  bbls.  of  potash,  each 
weighing  £91  lb.  gross,  tare  lOfb.  per  cwt.  ? 

Ms.  "1281/6.  Ooz. 
.  In  £5  barrels  of  %s,  oacli  Scwt.   Iqr.  gross,  tare 
pi:r  cv»t.  I61b. ,  how  iiiucu  neat  wcichtr 

"./?'«.«?.  ABcwt.  Mia. 
5.  In  SScwt.  Sc^rs.  gross,  tare  201b.  per  cwt.   what 
neat  weight? 

G.'Iii  45c\yt.  Cqn;.  CIIj.  ^^rus^^  iiuo  bib.  pur  cwt,  how- 
much  neat  weigjit  r 

jJns.  4^cwt,  %r<;.  IT^lb. 

r.  Vrhat  istiic  value  of  ilie  neat  weight  of  B  hhds.  of 
S.ugar,  at  ;S^,  54cts.  per  c^\t.  each  weighing  ICcwt.  Ic^^. 
I4lb.  gross,  tare  14lb.  per  cwt.  r 


Vv  hen  Trclt  is  allowed  witli  the  Tare. 

I.  Find  the  tare,  whicli  subtract  from  the  gi'oss,  and 
aHl  the  rcraainder  s'attie. 

S.  Divide  the  suttic  by  26,  and  the  quotient  will  bo  ihe 
trett,  which  subti-act  froin  the  ?iufile,  and  t'ne  remainder 
wil'l  be  tha  neat  wciglit. 

T.  In  a  i:o^^^hc:i:l  orsu;;ar,  weighing  iOcVv-t.  Iqr.  12ib. 
gross,  tare  I4lb.  ]«Cr  cwL  treU'4lb.' per  1 041b.*.  how 
much  neat  weight  ? 

*Tkis  is  th^  iveit  ailcwcd  in  London,     TUe'^reahtm  of 
dividing  hu  £6  U  Im'iuL^r.  Alh.  U  t,^^  of  104lb,  hit  if  Vt9 
Irett  is  at  anij  other  rate,  other  "pavtry  must  bz  falcon^  aC" 
O'cdin^  to  iha  veil p:  fro ;w$pdx  S'c. 


♦L'/lRB  ANP    THRTT. 


cwt.  qr.  lb. 
10     1     12 
4 

Or  thus 

cwt. 

14lbo4)10 

1 

qr. 
I 

1 

lb. 

12  gl'0S9. 

5  tare. 

41 

28 

26)9 

0 
1 

7  suttle, 
11  trett. 

sso 

8S 

An$.  8 

2 

24  neat 

4  =1)11  GO  gros^'.. 
^45  t:ir-. 

26)1015  suttle. 
39  trelt. 

Mtis,  9761b,  neuL 

2.  In  9  cwt.  2  qrs.  17  lb.  gross,  tare  41  lb.  trett  4  lb. 
per  104  lb.  how  irmch  neat^^       Ans.  Scwt.  Sqi^s.  QOlb. 

S.  In  15  chests  of  sugar,  weighing  1 1/  cwt.  21  lb.  gross, 
tare  173  lb.  trett  4  lb.  per  104,  how  many  cwt.  neat  ? 

Ms.  lllcivt.  22^6. 
4*  What  is  the  neat  weight  of  S  tierces  of  rice,  each 
weighing  4  cwt.  3  qrs.  14  lb.  gross,  tare  16  lb.  per  cwt. 
and  allowing  trett  as  usual  .^ 

.dns.  12ewt.  Oqrs.  6lb. 

5.  In  25  barrels  of  figs,  eacli  84  ib.  gross,  tare  12  lb. 
per  cwt.  trett  4  lb.  per  104  Ib.^  how  many  pounds  neftt? 

\^ns.  18034- 

6.  What  is  the  value  of  t}>e  neat  weight  of  4  barrels 
of  Spanish  Tobacco ;  nunibers,  weights,  and  allowanfres 
as  follows,  at  9 id.  per  pound  ? 

cwt,  qrs.  lb. 
No  1    Gross    1     2     irr^ 


1     0    25  1      Tare  16  lb.  per  cwt. 
1     0    09  f    Trett  4  lb.  per  104  lb. 


S     2!^ 


*/lns..£]7  I6». .^ef. 


TARE     iND    TRETT  ilD 

CASE  V. 

When  Tare, Tr^tt,  aiid  Cloft'  aie  allowed  : 
Deduct  the  tare  and  trett  as  before,  and  divide  the  sut- 
Hq  by  168  (because  2  lb.  is  the  -j}^  of  3  cwt.)  the  quo- 
tient will  be  the  clolT,  w  hich  subtract  from  the  suttle,  and 
the  remainder  will  be  the  neat  weight. 

2:XAMPLES. 

1.  Jn  5  hogsheads  of  Tobacco,  each  weighing  13  cwt. 
S  qr§.  23  lb.  gross,  tare  107  lb.  per  hogshead,  trett  4  lb. 
per  104  lb.  and  clofT  2  lb.  per  3  cwt.  as  usual  5  how  much 
neat. 

cwt,  qrs.  lb. 


00 

28 


443 

112 


1563  Ib.p-rossof  1  hiid. 


4689  whole  gross. 


10rx5«  321  tare. 

26)4368  suttle. 
168  tvQit 

168)4200  suttle. 
25  cloli: 


Jins.  4173  neat  weight. 

S^  What  IS  tlie  neat  v/elght  of  26  cwt.  S  qrs.  20^).  gross 
tar9  52]b.  the  allowance  ofti'^Ut  and  cIorTasusu.-il  f 

dns.  neat^Z^cwt,  lqr::5lb.  loz,  iiearli/ ;  qtMUh^ 

further  fractions^ 


INTEREST. 

InIIiIUEST  is  of  two  kinds ;  Simple  and  CompOttad* 
SIMPLE  INTEREST. 

Simple  Interest  is  i}\Q  sum  paid  by  the  borrower  ta  the 
lender  for  the  use  of  money  lent ;  and  is  generally  at  A 
certain  rate  per  cent,  per  annum,  which  in  several  of  the 
United  States  is  fixed  by  law  at  6  per  cent,  per  annum  ; 
tliat  is,  6/.  for  the  use  of  lOOL  Or  6  dollars  for  the  use  of 
100  dollars  for  one  year,  &c. 

Princina!,  is  the  b'um  lent. 

Rate,  IS  t]\Q  sum  jifcr  cent,  agreed  o*n. 

Amoant,  is  flie  principal  and  interest  added  together. 

CASE   I. 

•n>  Slid  the  interest  Of  any  given  sUm  for  one  year. 

RtJLE. 

Multiply  the  principal  by  the  rate  pet  cent,  and  divide 
the  product  by  100  5  the  quotient  will  be  the  answer. 

EXAMPLES. 

1.  What  is  tlie  interest  of  39Z.  lis.  S^d.  for  one  year. 

at  6/.  per  cent,  per  annum  ? 

£.     5.     d. 
S9     11     8^ 
6 


21S7     10    S 


7\5() 

6|03 
4 


0|I2  JIiis,  r;i  Th  eri^ii, 

2.  What  is  thciiiUrcsl  of  536^  10s.  4.^-  lor  ajraf,  ai 
rje^cfiJit^  *vti3\  £U  IBs.  Qd. 


SIMPLE    INTERJiSn  l£l 

5:  VVI/at  is  tlic  interest  of  571?.  13s.  9d.  for  one  year, 
at  6/.  per  cent.  ?  dns,  £34  6s.  O^d. 

4.  What  is  the  interest  of  2?.  12s.  9^d.  for  a  year, at 
6?.  per  cent.  ?  dns.  £0  5s.  Qd. 


FEDERAL  MONEY. 

o.  Wliat  is  the  interest  of  468  dols.  4,5  cts.  for  one  year 
at  6  per  cent.  P  S   cts, 

468,  45 
6 


\  S&jlO,  rO==S28,  Wcls.  7m,  Ans. 

Here  I  cut  oil' the  two  right  hand  integers,  which  di- 
vide by  100 :  but  to  divide  federal  money  by  109,  yon 
need  only  call  the  dollars  so  many  cents,  and  the  inferior 
dsnominations  decimals  of  a  cent,  and  it  is  done. 

Therefore  you  may  multiply  the  principal  by  the  rate, 
and  place  the  separatiix  in  the  proauct,  as  in  multiplica 
^on  of  federal  money,  and  all  the  figures  at  tlie  left  ol 
tiie  separatrix,  will  be  the  interest  in  cents,  and  the  first 
figure  on  the  right  will  lie  mills,  and  the  others  decimals 
ofamiil,  as  in  the  follow!  m^5 

EKAMPLES. 

6.  Required  the  interest  of  135  dols.  £5  cts.  for  ayeat 
at  6  per  cent.  B  els 


year  at  5  per  cent 


or,  5n=^9rcts.  5hm.  Ms. 
^\  "What  is  i'^e  liitcrcst  of  436  doi.ui^s  for  one  year,  at 


135, 

G 

lie 
hrs 

5  i  cen 

Jlns. 
ts  for 

811, 

Ucrest 

50=.SS, 
cf  ID  da 
$^   ct:<. 
19.  51 

one 

6 

pi  SiMi'LIi    iKlERES,!'. 

ANOTHER  METHOD. 

Write  down  the  ^Iven  principal  in  cents,  whicli  imiUI- 
ply  by  t.ic  rate,  and  divide  bj  100  as  before,  and  you  will 
have  the  interest  for  a  year,  in  cents,  and  decimals  of  a 
cent,  as  follows : 

9.  What  is  ihe  Lilerestof  §73,  o5  cents  ibr  a  year,  at 
6  per  cent.  ? 

Priucinal  7365  cents. 
6 


.Ins,  44\,90cts,=r=:44i-^^c{s.  or  S4,41cf5.  9m. 
10.  Required  the  interest  of  ^85,  45cts.  for  a  year,  at 
7  per  cent.  ^  Cents. 

Principal  8545 


Jlns.  598,  13  cenfs^r=^^5fiocts.ihm. 
CASE  ir. 

'io  find  the  simple  interest  of  any  sum  of  money,  for  any 
number  of  years,  and  parts  of  a  year. 

GENERAL  RULE. 

1st.'  Find  tiie  interest  of  the  jj;iven  sum  for  one  year. 

2d.  Multiply  the  interest  of  one  year  by  the  g»-e!i 
number  of  years,  and  the  product  v,  ill  be  tl ic  answer  for 
til  at  time. 

3d.  IF  tiiere  be  parts  of  a  yea",  as  iiU)ritlLS  and  day.^, 
work  for  the  months  by  tlie  alicjuot  parts  of  a  fenvn  and 
for  the  days  by  tiie  Rrde  of  Tluee  Direct,  or  by  allowin;!; 
30  days  to  Uu^  ;.:;:!,.  and   taking  arfquot  parts  of  the 


^'  By  ri[l(;\ving  tlie  ni.  ..:'..  U)  wc  CO  days,  and  taking  aliquot 
jKirtsthrreof,  you  will  l:.avc  the  intercut  jif  any  ordiniv.'y  sum 
sulficirnliy.  exact  for  common  use;  hut  if  the  surn  be!  very 
jnrj^e,  yon  rrmv  ^vo-v. 

As  :a>:/  (iay.'i  :  \^  to  the  intcrefst  of  onf  y^ :-"  :  :  so  is  ti:;. 
^ivci!  i-umbc."  s!i'  :::iv3  •  to  the  iotcr:-     ' 


•  IMTLF    TKTEnKST. 


'\ri 


:.  What  is  tlic  interest  of  To/.  F.s.  4u.  lor  5  years  and 
£  months,  at  G/.  per  cent  per  aiiiuim  ? 


75     8      4 


4;5s  10    0 


G  I    tlnw.^-VA  10  0  IiiiCroL.t  Tor]  vear. 


20 


UJ|50 
()iOO 


9.^1  le  G  (ia.  ibr  5  years. 
OUT  do.  for  2  raonths. 


2.  What  is  the  iiiierrst  of  64  lioiian:,  53  cents,  for  2 
ycai'S,  5  months,  and  10  chr:,  ;it  5  per  cent.  .^ 


522,90    l!itcre*t  for  1  vcr.r  in  cental,  per 
S  "  rCare  L 


4  mo.    ^ 
X  mo.     i 


OCS,rO  tlo.  lur  3  years. 
107,63  do.  fur  4  mourns. 
2G'90  (h>.  (Vm-  1  ?nonth. 


lOdavs,^  8.9G  do.  for  10  days. 

.te.  1112,ir):=ll!^:cf<?.  o:- Sn,  1::<\  ^^Jf. 

o.  Vriiat  is  iiie  interest  of  TS9  dollar::-  tor  2  years,  at  6 
pt-rcent.  ?  Jlns.  S94,  GScis, 

4.  Of  37  dollars  50  cents  Uiv  1  years  at  G  percent. per 
annum  .^  *  jLis,  OOOr/'y.  or  S9 

5.  Ur  525  dollars  41  ccnt^,  i'ov  3  years  and  4  monthsj 
at  5  percent.?  *6f?2s.  ;['>'54,  HP.cts,  5m, 

G.  Of  S^:i5/.  1,^2.  3d.  fx'  5  years,  at  6  per  cent.  ? 

J^ns,  £97  33.?.  8^. 
r.  Of  174^  ICs.  6*1.   for  3  and  a  half  vears  at  6  per 
cent.?  Ans,  £3G  13.s. 

8.  Of  150/'    I6s.  ud.  for  ^  vears  and  7  months,  at  6  per 

nt.  ?  "  .^ns,  fAi  a<?.  rd 


9.  Of  i  dollar  for  12  years  at  5  per  cent.  ? 

Ans,  60cts, 
ID.  Of  215  dollars  34  cts.  for  4  and  a  half  years,  at  5 
and  a  brnf  per  cent.  ?  Jn$,  ^33,  9lcts>  6m. 

11.  What  is  tne  amount  of  3^4  dollars,  61  cents?  for5 
jcars  aiid  5  months,  at  6  p^^r  cent.  ? 

..2;2S.  S450,  lOc/5.  3-^W^i. 

12.  What  will  SCOO'.  amount  to  in  12  years  and  10 
months,  at  6  per  cent.  ?  ^Ins.  £5310. 

13.  Wliat  is  die  interest <?f  257^.  5s.  Id.  for  1  year  and 
5  quarters,  at  4  per  cent.  ?        ^flns.  £18  Os.  Id,  Sqrs. 

14.  What  is  tiic  interest  of  279  dollars,  87  cents  for  2 
jGiirij  and  a  half,  at  7  per  cent.  ]>i}v  annnr.i  ? 

^^  » A (.-.''.  |j»4S,  "cfTcts,  T^ui 

15.  Wiiat  will  £r9/.  K-.',  od.  amoiuit  t(>4n  3  years  and 
a  half  at  5-|per  cent.  !)er  aiui'uii  ?  x 

J?.ris,  £331  Is.  6d. 

16.  What  is  the  amount  orG41  dol^.  60  cts,  for  5  years 
and  o  (|uai'ters,  at  7  and  a  lialf  per  cent,  per  annum  ? 

17.  What  v/ill  TSCi  dols.  amount  to  at  6  per  cent,  in  § 
years,  7  months  and  12  days,  or  -gVj  of  a  year  if 

'dns.  '^975,  99cts. 

18.  Wlait  is  the  interest  of  1S25L  at  5  per  cent,  per 
annum,  from  March  4th,  1796.  to  March  29th,  1799,  (al- 
lowing the  year  to  contain  365  days  ?)        Ans,  £280. 

Note, — The  Rules  ibr  Simple  Interest  serve  also  to 
calculate  Commission,  Brokerage,  Insurance,  or  any  thing 

else  cstijnatcd  at  urafiiyar  ceuh 


COr.I?viISSION, 

IS  an  allowance  of  so  niucli  per  cent,  to  a  factor  or  cor- 
respondent abroad,  for  buying  and  scl!!ng  goods  for  Ids 
ejnployer 

EXAMPLES. 

1.  What  will  tlvc  commissiou  of  843.^  tOs.  come  to  at 
5  per  c^nt»  ? 


£,    s.  Or  thus, 

8&  10  £.    s. 


42j  17  10  v^w.9.  £42  5  6 

-       20 

SjSO 
12 

6|00  £42  3s.  ed, 

2.  Required  tlie  commission  on  964  dols.  90  cts.  at  2i 
percent.?  .^«5.  S21,  71ct9.^ 

3.  What  may  a  fiictor  tlemantl  on  1  ^  per  cent,  commls- 
sion,  for  lavinp;  out  S568  dollars  ?        .:i/i>'.  ^^2.  44c{s, 


lUlOKEIlAGE, 

IS  an  allowance  of  so  much  per  cent,  to  persons  assist- 
ing merchants,  or  factors,  in  purciiasHng  or  s*d ling  goods. 

EXAMTLES. 

1.  What  is  tlie  brokerage  of  750!.  8s.  4d.  vd  6s.  Sd. 
per  cent.  ? 

s.    (L 

Here  I  first  find  the  brokerage  at  1 
pound  per  cent,  imd  then  for  the 
given  rate,  v/hich  is  ^|  of  a  pouniV. 


£.     s. 

750     8 

d 
4 
1 

7,50     8 
20 

4 

10,08 
12 

.9.   fZ.  £.   S,   (?.  fiTS. 

G  S=zl)7  10   I 


.i?2S.  £2  10  0  H 

-1,00 

2.  What  is  iAc.  brokerage  upon  4125  dols.  at  i  or  vo 
<^ints  per  cent.  ?  .fiua.  gSO,  93c^v.  7Ai». 

3.  if  a  broker  sells  goods  to  tiie  amount  of  5000  dwls. 
what  is  h's  demand  nt  Go  cts.  per  ccui.  ? 

>  *'2,^s.  g32,  50ciS 


it  is  tAc  brokerage  ui: 


4.  Wfiat  niiiY  abrolcer  dcniandj  when  hos«^Ilsgooas  i:»> 
the.  value  of  oO'iL  i  7*.  iOd.  and  I  allow  him  1  ^  per  cent.  ? 


INSURANCE, 

IS  a  premium  ai  so  luiich  per  cent,  allowed  to  persons 
and  oiTices,  for  rnakiiig  good  tlie  loss  of  ships,  houses,  mar- 
chandr/.e,  &c.  whicli  miiy  iiapperi  from  storms,  lire,  &c 

ICXAM?LES. 

1.  What  IS  the  iusiirance  of  T'25L  8s.  I'Od,  at  12^  per 
c&rit,  ?  */2n.*^.  rOu  13s.  7i^. 

2.  Whni  h  the  insurance  of  an  East-fmna  ship  and 
cargo,  valued  at  12o4£5  dollars,  at  15}^  ?)er  cent.  ? 

5.  A  maifs  house  estiinated  at  3500  dollars,  was  insu- 
red against  fire,  for  i|  per  cent,  a  year  :  v.hat  insurance 
did  lie  annually  pay  ^  ^flns.  S6l,  ^5ots: 


SHORT  PRACTICAL  RULES,     . 

Jun^  calczilfiliv^'  iHlerest  citQjier  cent,  either  for  monthsif 
or  montlis  and  days, 

L  FOR  STERLING  MONEY- 

RULl!:. 

1.  If  the  principal  coni^lsts  of  pounds  only,  cut  oiTtk* 
nnit  figure,  and  as  it  then  stands  it  will  be  the  interest  for 
one  month '^  in  shillings  and  decimal  parts. 

2.  If  the  principal  consists  of  pounds,  shillings,  &c.  re 
d nee  it  to  its  dcci»^-ml  value;  then  remove  the  decimal 
point  one  |)iace,  or  figure,  furtker  towards  the  left  hand, 
and  as  tl^e  decimal  tmn  stands,  it  will  shew  the  interest 
for  one  month,  iu  shillings,  and  decimals  of  a  shilling. 

KXAMPLILS. 

t.  Requue<!  tUe  interest  of  541  for  seven  montlis  and  i 

ten  o'ays,  at;  G  i>er  cent 


10  ihjs=};5,4  Interest  for  oneHfiontlv 


57,8  ditto  i\)Y  7  months. 
1,8  ditto  lor  10  days. 

Ms.  39,6  shi!Iings=£l  195.  TfitL 
12 


2,   What  IS  the  interest  ol'  42f.  10s.  for  11  month?,  at  6 
j;;D?  cent.  ? 

/;.   .^.     £. 

4'2  10  «=:  4£^  deciuuil  value.' 
Thcrorof  e  4^25  sbiin!';?;s  interest  for  1  n^oiith. 
'll 

-    *^is.  46,75  Interest  for  11  mo.  p=- ,"3    6    9 

Q,  llequiriBfl  the  interest  of  94L  7s.  Gxl.  for  one  yer.r, 
d'/e  mon^s  and  a  half,,  at  %  per  cent,  per  ;ui;.i;m. 

Ans,  £8  5s.  Id,  ry,r>qvs. 

4.  Whiit  \s  the  interest  of  121.  18s.  for  one  third  of  a 
KCSith,  at  6  per  cent.  .^         .  J^ns.  5;l6d, 


II.  FOK  FEDERAL  MONEY 

RULE. 

i.  Divide  the  principal  by  £,  placing  the  soparatn::  a% 
usAial,  and  the  quotient  will  be  the  interest  for  one  month 
in  cents,  and  decimals  cf  a  cent  5  that  I^,  the  figui*es  at 
the  left  of  the  separatrix  v/ili  be  cents,  and  those  on  the 
rj-rht,  tiecimalsoi  a  cent.  I 

9.  -VHuItiplythe  inler^'st  of  one  nuM?t'»  by  the  given  nunv 
?>.;-  >f  tnontJui,  or  month j*,iir»(l  decini;'!  parts  tiiercof,  or  for 
V  e  dr^ys  Uikc  the  even  v.:v:l6  of  a  month,  <fcc. 


EXAHPLES. 

1.  What  is  the^tercst  of  341  dols.  52  cts.  for  7h  montha  i 
£)S41,52 

Or  tlius,  170,76  Int.  for  1  monihi 

170,76  Int.  for  1  month.         x7,5  months 

r^  


85580 


1195,32  do.  for  7  mo.  119532 

B5,SS  do.  for  ^  mo. g  cfs.?7i. 

1280,700c^s.  =12,80  7 

1^80/0  ^ns.  1280,rcf.s.=S12,  SOc^s.  7«i. 
2.  Required'tKe  interest  of  10  dols.  44  cts.  for  S  v&AXS^ 
5  months  and  10  daj|, 

2)10,44 


10  (lays~I)     5,22  Interest  for  1  month. 
41  months. 


5,23  f 
208,8 

214,02  ditto  for  41  months. 
1,74  ditto  for  10  days. 

rZ15,76cts.     Jlns,==^9.,  IScts.  Tw.-f* 
5.  Wliat  Is  the  interest  of  542  dollars  for  11  months  ? 

The  i  is  171  Interest  for  one  monfc 
11 

Jns,  ldSlci5.==St8,  Slcts. 

NoTB.^ — To  fuid  the  interest  of  any  sum  for  2  months 
at  6  per  cent,  yon  need  only  call  the  dollars  so  many  cents, 
.ind  the  inferior  denominations  decimals  of  a  cent,  and  it 
is  done  :  Thus,  the  interest  of  100  dollars  for  two  mouths, 
IS  100  cents,  or  I  dollar;  and  S25,  40  cts.  is  25  cts.  4  m, 
&c.  which  gives  the  following 

RULE  II. 

Multiply  the  principal  by  half  the  number  of  moijthf, 
and  tiic  product  will  shew  the  intere^for  the  given  time, 
in  ccr.ts  and  dcoinials  of  a  cent;  as  above. 


■.:m?:.e  ixterkst. 

EXAMPLIi^S.  -^i 

Mtquiredtueiaterestof  316  dollars  for  1  year  and 
nionttis.  1 1  =half  the  number  of  ino. 

Ans,  S4r6cis. =S34,  76cts. 
\  What  is  the  interest  of  564  dols.  $25cts.  for  4  months? 
S    cts. 

2  half  the  months. 


'IL  IVhcn  Iho  pijncipai  Is  given  in  feilen'vi  monev,  at 
rr'^ercent.  tofiiiu  iiuw  niuchthc^  iiiontlvv  interest  m'iU be 
lit  New-England^  CSX.  cui-:-eiicj. 

RULE. 

?.Inltiply  the  given  principal  liy  ,03  and  tlie  product 
v.iW  be  the  interest  for  one  inoniii,  in  sliillings  and  deci- 
mal parts  of  a  shilling. 

EIIAMPLES. 

: .  What  IS  tlie  interest  of  325  dols.  for  11  months  ? 

,03 

9,73  shil.  int.  Cor  1  month. 

XH  montlis. 

.5ii.s\  107S5s.~£5  7s.  Sd, 
-  Vvhat  is  tlie  iiiterest  in  New-Kngland  currency,  of 

.    7^'s.  68  ct?^  for  5  montiis? 

r-rincipal  GI.GS  doh, 

.950*4.  Intercut  for  one  month. 

5 

0  UIZ^O 


180  ,IM-LE    l^rrKRK**'". 

IV.  When  the  ];rincipru  is  given  in  pounds,  shiiiings, 
&:c.  New-England  currency,  nt  G  per  cent,  to  find  how 
muck  iYiQ  man  tidy  interest  will  be  hi  federal  money. 

RULE. 

Multiply  the  pounds,  &c.  by  5,  and  divide  that  pro- 
duct by  3,  the  quotient  will  be  the  interest  for  one  months 
in  cents,  and  decimals  of  a  cent,  &c. 

EXAMFLES. 

1.  A  note  for  £411  Nev/ -England  currency  has  been 
on  interest  one  nioiitli ;  how  much  is  the  intea^Bt  thereof 
in  fedend  nionev  ^  f\. 


sm55 

2,  Required  tlie  interest  of  59^.  18s.  N.  E  currency 
for  7  months  t  £, 

S9p9  decnnal  value 
5 

3)199,5 

Interest  for  1  nio.     GG^5  cents. 


r.-^ 


V.  When  tlie  principal  is  given  ia  !Nf^;v -England  anil 
Vir2:inia  currency,  at  G  per  ceutt  to  find  the  mterest  for 
a 


V.  Vvhen  the  principal  is  given  la  iNf^;v-r^ngiand 
irginia  currency,  at  G  per  ceutt  to  find  the  mteres 
yeai*,  in  doH  arrtHiiills,  by  inspec^LOin__ 


koi.^: 


Since  the  inlerer^t  of  a  year  will  he  ju^t  so  many  cents 
as  the  given  principal  contains  shillirtj^s,  tiierefore,  write 
down  the  shillings  and  calltliem  cents.  Vend  the  pence  ifl 
tlie  principal  made  less  by  1  if  they  exceed  3,  or  by    ^ 
whan  tlwy  exceed  9,  will  *be  the  mills,  very  nearly. , 


SIMPLE    INTEREST*  131 

EXAMPLES. 

1 .  What  is  iht  interest  of  2/.  5s.  for  a  year  at  6  per  ct.? 

£2  5 s,=z45s.  Interest  45cts.  the  Jiii&tter, 

2.  llequireilthe  interest  of  IQOL  for  a'year  at  6  per  ct? 

£100»=2000h\  Interest  2Q0Qcts.==^20  Ans. 
S.  Of  Srs.  6d.  for  a  year  .^ 

dns,  27s.  is  SlTcts.  and  6d,  is  5  mills. 
4.  Required  the  interest  of  51.  10s.  lid.  for  a  year  ? 

£5  10s.=UOs.  Interest  IWcts.^^i,  lOcts.  Qnu 
1 1  pence — 2  ;}cr  rn le  ha ves  9  =  9 

Jlns.  SI,  10       9 

VI.  'i'o  compute  tiie  interest  on  any  iicte  or  obligatioV, 
when  there  are  payments  in  part,  or  indorsements. 
RULE. 

1.  ^ift^  tl^e  amount  of  the  whole  principal  for  the  whole 
time/jK 

2.  C^t  the  interest  on  the  several  payments,  from  the 
time  tiiey  were  paid,  to  the  time  of  settlement,  and  find 
their  amount;  and  lastly  deduct  the  amount  of  the  seve- 
ral payments,  from  the  amount  of  the  principal. 

EXAMPLES. 

Suppose  a  bond  or  note  dated  April  17, 1793,  was  given 
for  675  dollars,  interest  at  6  per  cent,  and  there  were 
payments  indorsed  upon  it  as  follows,  viz. 

First  payment,  148  doHars,  May  7,  1794. 

Second  payment,  341  dols.  August  17,  1796. 

Third  payiiient,  99  dols.  Jan.  2,  1798.  I  demand  how 
much  remains  due  en  said  note,  the  17th  of  June,  1798  ? 

118,  00  first  payment,  May  7,  1794.         Yr.  mo. 
o6,  50  interest  up  to—June   17,  179S.=4      H 

184,  50  Umount. 


341,  CO  second  payment,  Aug.  17,  1796.  Yr.mo. 
57,  51   Interest  t\) June"  17,  1798.  =1     10 

5/8,  61  amount ^ 
-. —  '  Carried  ovei^ 


99,  00  tJiud  pavinent,  January  2,  179S. 
2,  72  Intere.st" to—June  17,  1798.«5ifito. 

lOi.  r£  aiuount. 


1K»>. 


184,  501 

578,  51  I  several  amounts. 
101,  r2j 

664,  rs  total  amount  of  payments. 

€75,  00  notci,  (lat^d  AiirU  IT,  irH:-, 
^09,  25  Inbvest  to— June  IT,  iTNo. 

884,  25  anion  lit  of  tiie  note. 
664,  73  amount  cf  pnyirsents. 

8219,  52  HMiialns  i\\\<t.  on  the  nolo,  JuiKi  17Jf_ 
St,  On  the  i6th  of  JiDiunry,  1795. 1  lent  James  faywell 
^00  dollars,  on  interest  at  6  per  cent,  ^vliicli  I  received 
bads  in  the  follovvini;-  partial  payniei;!?.,  us  iiTidcr,  v;/.. 
Istof  April,  179G        -      '-        ~        -        S  ^'JO 
16th  of  July,  17?u  -         -         -  -    40O 

1st  of  Sept."    1798        -        -         -         -  CO 

How  stands  ih^  balance  bctv.ecn  i:s,  on  the  16th  Nu» 
YCmbcr,  1800  ?  Ans,  due  to  me  S63,  IBcls. 

3,    A  IllOMISSOllY    NOTE.   VIZ. 

£Cr2  105,  J.'*ew-Lo}}xluiU  .fjpnt  4,  1797. 

Oa  demand  I  proiuise  to  pay  Timothy  Cureiul,  bixly- 
two  pounds,  t^n  shillings,  and  i'licrest  at  6  |:er  cent,  per 
annum,  till  paid;  value  i(:celved. 
John  Staxijv,  I'KTKR  PAYWELL. 

RiCIIAUD  Tf-stis. 

Tiidonri^ient^*  £*   ■§» 

1st.  Heccivcd  iu  piirt  of  liie  above  note,  Sep* 

tembcr  4,  1799,  50    0 

And  payment  June  4,  ISOO,  12  10 

IIow«u;ch  rcrnuiiUi  due  oa  .-jaid  note,  i.lu>  {<Ktydi  day  ft 
Oeccmber,  1800  ?  1*.    .*^-.   f*^ 


SIMILE   INlEHESr.  135 

KOTK.  jtV/5  preceding  Bide^  by  customh  rendered  so 
popular,  and  so  much  jiractised  and  esteemed  hj  many  on 
account  of  its  being  simple  and  concise^^  thati  have  given 
it  a  place :  it  may  answer  for  short  periods  of  time^  but  in 
a  lo9^  coursa  oj  years  it  will  be  found  to  be  very  erro- 
neouff, 

Mhoiigh  this  method  seems  at  first  vieiv  to  leuponihe 
ground  of  simple  intevcf^t*  yet  upon  a  little  attention  the 
jollowing  objection  ivill  be  found  most  clearly  to  lie  against 
it^vi::^,  that  the  interest  will^  in  a  course  (f  yvars,  covti- 
ptsldy  expunge,  or  as  it  r,iay  be  said  eat  up  tfi£  debt.  For 
an  explanation  of  Ihis^  take  the  following 

A  lends  B  100  doHars,  at  6  per  ccirt.  iiitcrest,  and 
takes  his  note  of  hand ;  B  docs  no  more  tluin  puj  A  at 
every  year's  cad  6  dollars,  (which  is  then  justlv  due  to 
Bfortao  use  of  his  money)  u:ad  has  it  endorsed  on  his 
gjote.  At  tliO  end  of  10  years  13  takes  up  his  note,  aii'J 
tlie  sUm  };e  has  to  pay  is  reckoned  thus  :  The  principal 
lOOdallarSj  on  interest  10  years  amounts  to  160  dolhu-s ; 
there  are  nine  endorsements  of  G  jh')![ars  each,  upon 
"which  tlic  debtor  claims  iiiierest^  one  for  9  ye?: rs,  the 
§ccond  for  8  years,  the  third  lor  7  years,  and  so  do\vn  to 
the  time  of  settlement:  tno  \YiioIe  a!nount  of  ih-c  several 
cndorscmcnis  and  their  intei'cst,  (as  any  one  can  see  by 
casting  n)  is  S^O,  20  cts.  this  subtracted  from  1(30  dol.-^. 
the  amount  of  the  debt  leaves  in  iiivour  of  ilio  creditor, 
389,  40  els.  or  iglO,  20  cts.  less  tl^an  the  original  princl- 
]3al,  of  which  he  has  nat  rctcived  a  cent,  but  only  its  an- 
nual interest. 

If  the  same  note  should  He  '20  years  in  the  same  v/ny, 
3  M'onid  owe  but  27  dols.  CO  cts.  without  paying  xhi:^ 
least  fraction  of  die  100  dollars  borrowed. 

Extend  it  to  28  years;  and  A  tlic  creditor  v.-rndd  fa!I 
in  debt  to  15  withoi'.t  leceivin^  a  cent  of  llie  100  dollars 
which  he  lent  him.  Sco  a  better  Rule  in  Bhinna  hittr- 
cat  by  ^Decimals,  pag&  I*!"?. 


io4  trj'j^VQtjsD  INTEREST, 

eOMPOUND  INTEREST, 

is  when  the  interest  is  added  to  the  pnncipal,  at  tlie  end 
of  t}^  jear^  and  on  i^.  amount  the  interest  cast  for  anotli- 
er  year,  and  added  again,  and  soon  :  this  is  called  Inter- 
e»it  upon  Interest. 

RULE. 

Find  tliQ  interest  for  a  year,  and  add  it  to  the  principal, 
\\  luch  call  the  amount  for  tlic  first  year ;  find  tlie  interest 
of  tins  amount,  vAnch  add  as  before,  for  the  amount  of  the 
second,  and  so  on  ibr  any  number  of  years  required. 
Subtract  the  original  principal  from  the  last  amonnt,  and 
the  remainder  will  be  the  Compound  Interest  foi*  tli^ 
whole  time. 

1.  Requircd  tlic  amount  of  100  dollars  fbr  5  jelrs  at  6 
per  cent,  per  annum,  compound  interest  ? 

Q\'h.  S  cts. 

ist  Principal  100,00  Amount  IU65OO  for  I  year. 
2d   Principal  106,00  Amount  112,36  for  2  year:-;. 
Sd   Principal  112,36  Amount  119,10l6for  Syps.  .^/is. 

2.  What  is  the  amount  of  425  dollars,  for  4  years,  at  5 
?ier  cent,  per  annum,  compound  interest  P 

dns.  S516,  odds. 
S.  Wiiat  will  400^'.  amoinit  to,  in  4  yeais,  at  6  per  cent* 
per  annum,  compound  interest  .^  ^Ins,  £504  \9s,  9 id, 

4.  What  is  the  compound  interest  oi  150/.  10s.  for  3 
years,  at  6  per  ct.  per  annum  ?     dns.  ^j28  145.  llld,-]" 

5.  What  Ts  the  compound  interest  ol  5C0  dollars  for  4 
yearo,  at  Gper  cent,  per  annum  ?  Jlns.  gl31,238-f 

6.  Vv'hat  will  1000  dollars  amount  10  in  4  years,  at  T 
per  cent  per  annum,  compound  interest  ? 

.']ns,  S1310,  79c[s.  Gnu-h 

7.  What  is  i\\(^  amount  of  750  dollars  lor  4  years,  at  6 
per  cent,  per  annum,  compound  interest  ? 

Jins,  S046,  Sods.  r,7272i. 

3.  What  is  tiic  corapound  interest  of  876  doIs.OO  cts. 
for  3*A  ynuh  ^^^  C  p,cr  cent,  pef^aapum  ? 

dns,  S198,  S3ef5.-l| 


DISCOUNT, 

iS  an  allowaucc  made  for  llic  payir.cnl  of  aiiv  sum  of 
money  before  it  becomes  due :  or  upon  adva-ncing  ready 
moi>€y  for  notes,  bilis^  &c.  whicb-are  ])ayab]o  at  a  future 
day.  NVhat  remains  after  the  discount  is  deducted,  is  the 
present  >vorth,  or  such  a  sum  as,  if  put  to  interest,  would 
at  the  given  rate  and  lime,  rDnount  to  the  given  si;m  or 
tiebt.  

RULE. 

As  the  aiTiount  of  100/.  or  100  dollars,  at  the  given  rate 
and  time  :  is  to  the  inlcrcfit  of  100,  at  iho.  same  rate  and 
time  :  :  bolfj  tl*.e  v:ive'n  sn:=i  :  to  t!i^^  d.iscount. 

Subtract  the  di^V^ount  fro?n  the  given  sum,  and  the  re- 
mainder is  the  present  worth. 

Or — as  tbe  amount  of  K)0  :  h  to  ICO  :  :  so  is  the 
giv^^  sum  or  debt  :  to  the  present  worth. 

Puoor. — Find  the  amount  of  t'ao.  present  worth,  at  the 
given  rate  and  time,  and  if  the  work  is  ri^hU  that  will  be 
eyual  to  the  given  sum. 

KXAMTLr.S. 

1,  What  must  be  discounted  fm-  tha  ready  payment  of 
100  dollars,  due  a  vcar  lience  at  G  per  centi  a  year  ? 

'S       S         B      S  cts. 
As  lOG  :  6  :  :  100  :  o  66  ih^  answer. 
IOO5UO  given  sum. 
5,66  discount. 

S5-^S34  tricp'-escnt  worth. 

2.  What  sum  in  ready  money  v/iil  discliarge  a  debt  of 
925?.  due  1  year  and  8  montiis  hence,  at  G  per  cent,  r 

10  Interest  for  £0  montiis. 

110  Am-t.  £,       £.  £.       £.    s.  d. 

As  110  :  lUO  :  :  9^5  :  840  18  2+^tis. 
C.  What  is  the  preser.t  worth  of  GOO  dollai'S,  duQ  4 
years  hence,  at  5  per  cent,  t  Jna*  S5C0 

4.  What  is  tl;c  tliscouiit  of  ^T5L  10s.  for  10  months, 
at  €  Dcr  cent,  per.atinum  ?  diis,  £\S  Qs,  4^4 


i ":» o  ,-,  :>  M  \ 


5.  Bought  gO(,uls  ainounniig  to  615  clol:?.  T^>  ccutSj  r-^ 
tuonths  credit;  how  muck  ready  moPiej  jn.nst  1  pay,  dis- 
count at  4A  per  cent,  per  anninii?         "^       .'Lis,  gGOG. 

G.  Wliat  sum  of  ready  money  must  be  received  (or  a 
hill  of  900  doIIarSydue  TS  days  hence,  discount  at  6  per 
cent,  per  annum  ?  "  dns.  gBSS),  S'Zcts.  Sm, 

Norn. — When  sundry  sums  are  to  be  paid  at  different 
tiine-^,  find  the  Rebate  or  present  vrorth  of  each  particular 
[laymeat  separately,  and  when  so  found,  add  thcra  intot 
ciiesum. 

It  of  7oS/.  ih^  one  half  payable 

:.    -.  .    ...  ..  .        :!icr  hall  in  six  months  after  that, 

o.  ii  a  iegacy  is  iei'tine  of  2000  dollars,  of  which  500 
tlols.  arc  payable  in  6  niontiis,  800  dols.  payable  in  1  year, 
and  i'ae  re^t  at  the  end  of  5  years :  how  much  ready  moncj 
ought  1 1;;  recch  e  lor  i:?mi  legacy,  allowing  6  per  cent. 
discounts  'Jin?.  B1833,  Sfcfs.  4m. 


ANNIliTiES.c/ 

/\N  Annuity  h  a  sum  of  jnouey,  payable. every  year,  oi: 
for  a  certain  nuuiber  of  rears,  or  forever. 

V/hen  the  debtos*  kecp.^.  ike  annuity  in  Ids  o^vn  lumds?, 
beyond  the  time  of  nayin^nt,  it  is  said  to'De  in  arrears. 

The  sum  of  all  the  annuities  for  the  tiine  th.ey  have  been 
favborne,  together  wiui  the  intei'-snlue  on  caclu  i.>  called 
liie  amount. 

If  an  annuity  is  bongr.i  vii.  <m'  paid  all  at  once  at  the 
bcginnfng  of  the  lirst  ye.ir.  (.i.vi  price  which  is  paid  fi>r  it 
h  called  the  present  wonh. 

To  find  the  amount  of  an  aniiuii  v  at  siuinle  interest. 
■RULE.      ^ 

1    Find  iitm  interest  of  tiie  given  riunuivy  (or  1  year. 

2.  Aud  tlien  for  5,  3.  kc.  yc:irs,  up  io  the  given  time, 
less  1. 

o.  MultiJily  th(i  annuity  by  t'.'i  number  of  years  given 
and  iv^d  ili'j  "product  to  the  u'lu'^      '        '  aiul  tl^e  sum 


♦, 


:.  liau  annuity  ofTOLhe  forbornseO'  ycais,  what  will 
b«  due  fur  the  prindpal  ami  iiHeiest  at  fn^end  of  said 
term,  simple  interest  being  completed  at  5  pei*  cent,  per 
annum?  «  IV.     £.  s. 

•    let  Interest  of  7QL  at  S.^^cent.  for        1—  3  10 

W^  2—  r  0 

3—10  10 

4—14     0 

^2u,  Anr  ;»' yi'S.  ;ninuiij,at  TOt.  per  yr.  Is      350     0 

.fins.  £385     o 

?L,  A  IiouaC  being  let  upon  a  lease  of  7  years,  at  400 

dollai-s  per  annnn;.  and  the  rciit  Imh^  in  arrear  for  the 

\\\w\q  term,  I  demand  tlic  sum  du<»  at  th€  end  of  i\\Q  term, 

simple  interest  being  allowed  at  61.  percent,  per  annum  ? 

dns.  S5304. 


To  find  the  present  worth  of  an  annuity  at  simple  interest. 

RULE. 

Find  t]\G  present  v.-orth  of  eacl;  year  by  itself,  discouril- 
ing  from  tlic  time  it  falls  duo,  and  the  sum  of  nil  these 
present  worllis  will  be  the  preiiciit  worth  rcqrjred. 

EXAMPLES. 

I.  Wjiat  is  the  present  ^vorth  of  400  dols,  pci'  annum, 
to  continue  4  years,  at  6  per  cent,  peri*  annum  P 
106"^  3rr.35849  =  Pres.  worth  ot  1st  vr. 

^^  L  .  inn  .  .  Am  .  357.1428.5  == 2d  yr. 

118  r  3385983O0  —     Sd  vr. 

I24J  522,58064  =     -~- 4th  yr. 


JJiis,  S1396,C6503~=S1396,  Gct<i.    5m, 
2.  How  mucli  ^ircsent  money  is  oquivalvint  to  an  an- 
nuity ot  100  dollars,  to  continue  3  yeju-s ;  rebate  bein;» 
iiade  at  6  per  cent.  ?  Jlns,"  S268,  STcts,  Im.  * 

S.  What  IS  SOL  yearly  rent,  to  continue  5  years,  worth 
bi  reajy  mon^.  at  6L  per  cent.  ?    .fins,  /;S40  155.  -h 

12^ 


KOUATION  OF  PAYMENTS, 

is  nulling-  tiie  equated  time  to  paj  at  once,  several 
debts  due  at  difiTerciit  periods  of  timcj  so  that  no  loss  shall 
be  sustained  hj  either  party. 

RULE. 
Multiply  each  payment  by  its  time,  and  divide  the  sum 
of  the  several  products  by  the  whole  debt,  and  the  quotient 
will  be  the  equated  time  for  the  payment  of  thev/hole. 

EXAMPLES. 

1.  A  owes  B  580  dollars,  to  be  paid  as  follov^s — viz. 
100  dollars  in  6  months,  120  dollars  in  7  months,  and  160 
dollars  in  10  months  :  What  is  tlie  equated  time  for  the 
payment  of  the  whole  debt  ?  I 

100  X     6  ^     600  0 

1^0  X     r  ^^     840 

160  X  10  ^  1600 


7 


SSO  ;3040(8  monlhs.   An.. 

^.  Anierchantliath  owing  himSOOL  io  be  paid  as  fol* 
iOws  :  50Z.  at  2  months,  100/.  at  5  months,  and  the  rest  at 
8  months  ;  and  it  is  agieed  to  make  one  payment  of  the 
^vhole  ;  I  demand  i\\^  equated  time  ?    Ans,  6  mont7is, 

o.  F  owes  11  1000  dollars,  whereof  200  dollars  is  to  be 
paid  present,  400  dollars  at  o  months,  and  the  rest  at  15 
months,  but  they  agree  to  make  one  payment  of  the  whole ; 
1  demand  vrhen  that  time  must  be  ?       Jlns.  8  months. 

4.  A  merchajit  has  due  to  him  a  certain  sum  of  money, 
to  be  paid  one  sixth  at  2  months,  one  third  at  3  months, 
and  the  rest  at  6  months  ;  what  is  the  equated  time  for 
the  payment  of  the  whole  ?  dns.  4|  months. 


BARTER, 

Is  the  exchanging  of  one  commodity  for  anothei-,  and  di- 
rectB  merchants  and  tradeis  how  to  make  the  exchangei 
\>ithout  loss  to  either  party. 

RULE. 
Eind   the  value  of  the  commodity  whose  quantity  is 
gij.eii :  then  linji  wliat  quantity  of  t!ic  other  at  the  pro- 


n  A  JIT  K.r..  ^ 

fOp^*i  rale  can  be  bouglit  Tor  the  same  money,  and  it  gives 
(he  answer. 

EXAMPLES. 

1.  What,  quantity  of  flax  at  9  cts.  pei*  lb.  must  be  given 
in  barter  for  12lb.  of  indigo,  at  2  dols.  19  cts.  per  lb.  ^ 

12lb.  of  indigo  at  2  dols.  19  cts.  per  lb.  comes  t().?6 
dols.  28  cts.— therefore,  As  9  cts.  ;  lib.  :  :  2tv28cts.  : 
292  the  ansrver. 

2.  How  much  Vueat  at  1  dol.  25  cts.  a  bushel,  must  be 
given  in  barter  for  50  busliels  of  rye,  at  70  cts.  a  basliel  ? 

^^ns,  28  bushels, 
5.  How  much  rice  at  28s.  per  cwt.  must  be  bartered 
fi)r  3icwt.  of  raisins,  at  5d,  per  lb.  ? 

Ms.  5civt.  oqrs.  9^f|^y. 

4.  How  much  tea  at  4s.  9d.  per  lb.  must  be  given  in 
barter  for  78  gallons  of  brandy,  at  12s.  Sid.  per  gallon  ? 

Jhi3.  Q.inlb,  13-1^-0::;. 

5.  A  and  15  bartered  :  A  had  8dcwt.  of  sugar  at  12  cts 
per  lb.  for  which  B  gave  him  IScv.t.  of  flour;  what  was 
the  lh)ur  rated  at  pcV  lb.  ?  Jus,  diets, 

6.  B  delivered  S  hhds.  of  brandy,  at  Gs.  8d,  per  gallon, 
to  C,  for  12G  yds.  of  clolli,  what  wa?»  the  cloth  pe  '  yar;l  ^ 

7.  F)  ^ivesE  250  yards  of  il  •.!•/' 
fop  5191b.  of  pepper;  what    :;::::. 

iutcr  lb.  ?  .  ;" 

8.  A  and  B  bartereu  :  A  had  ^ixwi- 
per  cwt.  for  w'nich  B  gave  WiTi  20/.  in 

rest  in  sugar,  at  8d.  \x^v  ib. :  l  demand   .iOv     • 
Bgavc  A  besides  the  20/.  .^  Jina  tdvt-  Qqy. 

9.  Two  farmers  bartered ;  A  had  iCO  bus-ieJ«  o:  ,.     .: 
at  n  dols.  per  bushel,  for  which  B  gave  him  100  busrie:- 
of  barley,  worth  fi5  cts.  per  bushel,  and  the  balance  in  (rat- 
al 40  cts.  per  bushel ;  v.hat  quantitv  of  oats  did  A  roct  r"- 

10.  A  hath  linen  clotli  worth  20d.  an  ell  ready  m  -.u-,  ; 
bul  in  barter  he  v;iU  have  2s,  B  liatli  broadcloth  v/ot  th 
14s.  6d.  per  yard  vendj  money,  at  v/hat  price  ougiit  33  to 
ratejtis  bix)adcloth  in  barter,  so  as  to  be  eqiiivnlent  to 
A^  bartering  pnr.c  ?  Jh:$,  \rs.  irJ,  Sf^cr:?, 


A  and  B  barter:  A  nath  145  gallons  of  brand  j  &t 

jL  £0  cts.  per  gallon  ready  money,  but  in  barter  lie 

.ill  have  1  del.  35  cts,  per  gallon  ;  B  has  linen  at  58  cts.' 

per  yard  ready  money ;  hov/  must  B  sell  liis  linen  per 

yard  in  proportion  to  A's  bartering  price,  and  how  many 

yards  are  equal  to  A's  brandy  ? 

Jlns,  Bai'ter  price  of  B's  Jinen  is  65cts.  S^wi.  and  he 
must  give  A  500  yds,  for  his  brandy. 

12.  A  has  £25  yds.  of  shalloon,  at  £5.  ready  money,  per 
yard,  which  he  bralcrs  v/ith  B  at  £s,  5d.  per  yard,  takir 
mdi^oat  l£s.  Gd,  per  lb.  which  is  worth  kit  lOs,  how 
much  indigo  >vill  pay  for  tlie  shalloon  ;  and  vrho  gets  the 
best  bargain  ? 

*^ws.  4Si?&.  at  barter  price  will  pay  for  the  shalhjon, 
and  B  has  the  advantage  in  barter. 

Value  of  A's  cloth  at  cash  price,  is  /:££     10 

Value  o(4Silk  of  indigo,  at  lOs.  per  lb.         '^£i     15 

B  gets  tlie  best  bargain  by     £  0     15 

Is  a  rule  by  which  merchants  and  ti-aders  discover  their 
profit  or  loss  in  buying  and  selling  their  '^oods  :  it  also  in- 
structs them  how  to  rise  or  fall  in  the  price  of  their  good^, 
so  as  to  gain  or  lose  so  much  per  cent,  or  otherwise. 
Questions  in  this  mle  are  answered  by  the  Rule  of  Thr(?e. 

EXAMPLES. 

1.  Bought  a  piece  of  cloth  containing  05  yards,  ibr 
191  dols.  £5  cts.  and  sold  the  same  at  2  tlols.  81  cts.  per 
yard;  wliatis  the  profit  upon  the  whole  piece  ? 

Ms,  S4r,  eocts. 

2.  Bought  12^  cv;t.  of  rice,  at  3  dols.  45  cts.  a  cwt. 
and  sold  it  again  at  4  cts.  a  pound ;  what  was  i]\e  whole 
gain  ?  ^iis.  S12,  STcts.  5m 

3-  Bought  1 1  cwt.  of  sugar,  at  64d.  per  lb.  but  could 
not  sell  it  again  for  any  more  than  2l.  iGs.  per  cv.t. ;  did 
1  gain  or  lose  by  my  bargain  r  .Ins,  Lost,  £2.  lis.  Ad, 

4.  Bou^lit  44  lb.  of  tea  for  6/.  It-s.  and  sokl  it  again  for 
S?.  lOs.  6d. ;  vvliat  v/us  the  profit  on  each  pound  r 

Ms,  to  id. 


iOSS   AND    GAIN  l^; 

o*  Kougkt  a  had.  of  molasses  containing  119  gallons, 
f.t  52  cts.  per  gallon;  paid  jfor  carting  the  same  1  dollar 
25  cents,  and  by  accident  9  gallons  leaked  out ;  at  what 
Fate  must  I  sell  the  remainder  per  gallon,  to  gain  13  dol 
lare  in  the  whole  ?  Ms.  69cts.  2m.-|- 

II.  To  know  what  is  gained  or  lost  per  cent. 
RULE. 

First  see  what  the  gain  or  loss  is  by  subtiiaction ;  then 
As  the  price  it  cost  :  is  to  the  gain  or  loss  i  :  so  is  1 00^ 
or  SI 00,  to  the  gain  or  loss  per  cent.  ^ 

EXAMPLES. 

1 .  If  I  buy  Irish  linen  at  2s.  per  yard,  and  sell  it  again 
at  2s.  8(1.  per  vard  ;  what  do  1  gain  per  cent,  or  in  laying 
out  lOOt.r       As  :  2.«?.  8^.  :  :  lOOL  :  £35  6s.  8c?.  Ans, 

2.  If  I  buy  broad«loth  at  S  dols.  44  cts.  per  yard,  and 
sell  it  again  at  4  dols.  80  cts.  per  yard  5  what  do  I  gain 
per  cent,  or  in  laying  out  100  dollars  ?  % 

S  ctsr] 
Soldfai-4,   SOI  S  cfs.    cts.        g        g 

Cost       3,   44  ^  As  3,  44  ;  8S  :  :  100  :  ^^ 

I  Jins.  25  per  cent. 

Gained  per  yd.     86J 

S.  If  I  buy  a  cwt.  of  cotton  for  34  dols.  86  cts.  and  sell 
it  again  at  41  i  cts.  per  lb.  what  do  I  gain  or  lose,  and 
what  per  cent.  ?  S  cts. 

1  cwt.  at  41  ids*  per  lb.  comes  to  46,48 
Prime  cost  34.86 


Gained  in  tlie  gross,  Sn,62 
xis  54,86  :  11,62  :  :  100  :  SS^  ^Ans.  53-]-  j^er  cent. 

4.  Bought  sugar  at  S^d.  pQ4'  lb.  and  sold  it  again  at  4Z. 
ITs.  per  cwt.  wliat  did  1  gam  per  cent.  ? 

Jus.  £25  195.  did. 

5,  If  i  buy  12  hhds.  of  wine  for  204'.  and  sell  the  same 
again  at  14^  17s.  Cd.  perhhd.  do  I  gain  or  lose,  and. what 
\yex  cent.  ?.  Jlns.  I  lose  12^^  per  cent. 

().  At  ^l{\.  profit  in  a  shilling;,  how  much  percent  ? 

Ms.  r\9.  105. 


5  i^w  LOSS    AN'S)     GArV. 

r.  At  25  cts.  prolU  in  a  d-]!iir,  how  iuuv^h  por  ccnf,  ? 

.^•?-?.<>'.  £5  ;;??•  cent. 

Note. — When  goods  are  l>ou<^!it  or  gold  on  credit,  you 
must  calculate  (by"  discount)  tlie  present  worth  of  tlieir 
price,  in  order  to  ibid  your  tiue  guia  or  loss,  k^cc. 

k:lamplks. 
1    Bouglit  iC4  yards  of  broadcloth,  at  148.  61],  per  yd. 
ready  money,  and  sold  the  same  again  for  154/.  lOs.'on 

6  months  credit;  what  did  I  «;;;tin  by  the  whole;  allow- 
ing discount  at  6  per  cent,  a  vear  ? 

As  1j3  :  100  :  :  1j4  10  :  Iju     0  present  worth. 

I  iS  IB  prime  cost. 

Gained  /jSl     2  Answer, 
2.  If  I  buy  ciotli  at  4  doLs.  lt>  cts.  per  yard,  on  eigi»t 
montlis  credit,  and  ^ell  it  ai^ain  at  3  dols.  90  cts.  per  yd. 
ready  money,  what  do  I  lose  per  cent,  allowing  6  per  cent. 
discount  on  the  purchase  price  ?         Jliis,  2^  per  cent. 


III.  To  kg^w  how  a  commodity  must  be  sold,  to  gain 
m*  kkse  so  much  per  cent. 

RL'LE. 
As  100  :  is  to  the  purchase  price  :  :  so  it»  lOOi^.  or 
TOO  dols*  with  the  profit  add^d,*  or  loss  subtracted  :  to, 
the  selling  price. 

EXAM?Lr.?>. 

1.  If  I  buylnsh  linen  at  2s.  3d.  per  yard  ;  now  r^*i 
I  sell  it  per  yard  to  g-ihi  £5  p^r  cent,  v 

As  lOOZ.  :  2s.  SJ.  :  :  125/.  to  2f.  9d,  Sfjrs.  Jins> 

2.  If  I  buy  Rum  at  1  del.  5  cts.  per  gallon;  how  must 
I  sell  itpergalk'jn  to  gain  SO  per  cent.  ? 

As  SlOO  :  Bl,05  :  :  B130  :  Sl,SG.3cis.  j]}is. 
S.  If  tea  cost  54  cents  per  lb. ;  how  must  it  be  sold  p^r 

lb.  to  lose  ^Q.l  per  cent,  r 

As  SlOO  :  54  cts.  :  :  ^Sr,  50  cts.  :  4rcfs.  Stlm.Jns, 
4,  Bouglit  Cioth  ITs.  Ch\,  per  yard,  which  not  provin** 

so  ^od  as  I  expected,  1  am  obliged  to  lose  15  per  cent. 

hf^ij  how  must  I  sell  it  per  vard  .^      ^^ns.  \A^.  IQ-'ii',    • 


5.  If  11  c\vt.  1  qri£.5  lb.  of  sugar  cost  126  dols.  50  cts. 
kow  must  it  be  sold  per  lb.  to  gain  SO  per  cent.  ? 

dns,  IScf.s.  Siru 

6.  Bouglit  90  gallons  of  wine  at  1  dol.  20  cts,  oei  oali. 
but  by  accident  10  gallons  leaked  out,  at  what  rate  i^cvit  i 
sell  the  remainder  per  gallon  to  gain  upon  i'ne  whole  prime 
cost,attlie  rate  of  12^  per  cent.  ?  Jins,  gl,  6icts,  ^'fjfn. 

IV.  When  there  is  gained  or  lost  per  cent,  to  knoy. 
»  what  the  commodity  cost. 

RULE. 
As  100^.  or  100  dols.  with  the  gain  |>er  cent,  added,  \}t 
loss  per  cent,  subtracted,  is  to  the  price  5  so  is  100  to  the 
prime  cost. 

EXAMPLES. 

1.  If  a  yard  of  cloth  be  sold  at  14s.  Td.  and  thei-eis 
gained  16/.  ISs. 4d.  per  cent. 5  what  did  the  j&xd  coat? 

£.    s.    d.     s.  d.         £. 
As    116  15  4  :  14  7  :  :  100  to  12^.  6d.  Jns, 

2.  By  selling  broadcloth  at  3  dols.  25  cts.  per  yard,  1 
lose  at  the  rate  of  20  j^er  cent. ;  whut  is  t!  -^  prime  cjat  oi 
said  cloth  per  yard  ?  Jns.  g4,  Qticis.  ^.h'm, 

S.  If40  lb.  of  chocolate  be  sold  at  25  cts.  psr  lb.  arxd  I 
gain  9  per  cent.  5  what  did  tiie  whole  cost  me  ? 

Jus.  S9,  I7cts.  4m.-i- 
4.  Bought  5  cwt.  of  sugar,  and  sold  it  again  at  12ceftt3 
per  lb.  by  which  I  gained  at  tlie  rate  of  25^  per  cent  ^ 
what  did  the  sugar  cost  me  per  cwt. 

Jhis,  SiO,  70c4s,9m.-t 

V.  If  by  wares  sold  at  a  ^ivcn  ral»  tSwe  U  ^  ?ai«jh 
gained  or  lost  per  cent.  t«  km>w  what  would  be  ^iWf^  w 
lost  per  cent,  if  sold  at  another  rat*. 

RULE. 

As  the  first  price  :  is  to  lOOZ.  or  100  dols.  wdth  the  f  refit 
per  ccjit.  added,  or  loss  per  cent,  subtracted  :  :  so  is  tb€ 
other  price  :  to  the  gain  or  loss  per  cout.  attJ^e  other  rate. 

N.  B.  If  ycur  answer  exceed  ICO/,  or  00  dols.  the 
•xccss  is  you:*  jjaln  per  cent. ;  but  if  It  N  ^ss  than  100, 
ftit  deficiency  13  flic  Io*s  percent 


144  FELLOWSUir. 

EXAMPLES. 

1.  If  I  sell  cloth  at  5s.  per  yd.  and  thereby  gain  15  per 
cent  what  shall  i  gain  per  cent,  if  I  sell  it  at  6s.  per  yard  ? 

6-.        £,         s,      £. 

As  5  ."  llo  :  ;  6  :  138  Jns,  gaimd  SS  per  cent 

£.  Li'  I  retail  ruiii  at  1  dollar  50  cents  per  gallon  and 
thereby  gain  £5  per  cent,  what  sliall  i  gain  or  lose  per 
cent,  if  1  sell  it  at  1  dol.  Sets,  per  gallon  ? 

S  cts.     S        S  cts,     S 

1,50  :  125  :  :  1,08  :  90  Ans.  I  shall  lose  10  per  cent. 

5,  If  1  sell  a  cwt.  of  sugar  fbr  S  dollars,  and  thereby 
lose  IS  per  cent,  what  shall  I  gain  or  lose  per  cent.  HI 
sell  4  cwt.  of  the  same  sugar  for  36  dollars  ? 

dnt^.  Hose  vnly  I  per  cenK 

4.  I  sold  a  watch  lor  i7"L  Is,  5d.  and  by  so  hieing  bst 
15  per  cent,  whereas  1  ojglit  in  trading  to  jiave  cleared 
20  per  cent:  j  how  much  was  it  sold  under  its  real  \a!ae  ? 

£  •  £  •       £•  ^'  ^^* 

As  85  :  :  ido  :  20  1  8  the  pi'ime  cxyst 

100  :  ^20  13::  120  ;  24  2  0  the  real  rulut. 
Bold  ibr    IT  1  5 


£T  0  7  diisiver. 


FELLOWSHIP, 

Ife  a  rule  by  whicli  the  accorapfs  of  several  uiercliants  or 
other  persons,  trading  in  partnership,  are  so  adjusted, 
that  each  may  have  Ids  sliare  oi'  the  gain,  or  sustain  his 
aliare  of  the  loss,  in  proportion  to  his  share  of  the  joint 
itock. — Also  by  this  Rule  a  bankrupt'*s  estate  maybe  di- 
vifled  among  liis  creditoi-s,  &c. 

SINGLE   FELLOWSHIP, 

Is  when  the  several  shares  of  stock  are  continued  in 
trade  an  equal  term  of  tirae. 

RULE. 

As  tne  v.'hplc  stock  is  to  tlie  whole  giun  or  less  :  sal% 
each  man's  particrJir  stuck,  tohi»partiCiililr  sh.**"*^  'vVlV^ 


TKLLOWSUIF.  145 

Proof.— Add  all  the  particular  shares  of  tlie  gain  or 
loss  together,  and  if  it  be  right,  the  sum  will  be  equal  to 
the  whole  gain  or  loss. 

EXAMBLES. 

1.  Two  partners,  A  and  B,  join  their  stock  and  buy  a 
quantity  of  merchandize,  to  the  amount  of  820  dollars ; 
in  the  purchase  of  which  A  laid  out  S50  dollars,  and  B 
470  dolLars  5  tlie  commodity  being  sold,  they  find  their 
clear  gain  amounts  to  250  dols.  AVhat  is  each  person's 
share  of  the  gain  r 

A  put  in  550 
B 470 

A^  800  .  2.0  .  .  S^^O  :  106,7073+A's  share. 
AS  b^u  .  ^ju  .  .  ^^^Q  ,  145,2926+B's  share. 

Proof  249,9999+  =S250 

2.  Three  merchants  make  a  joint  stock  of  1200/.  of 
which  A  put  in  240/.  B  S60/.  and  C  600/.— and  by  trading 
they  gain  325/.  what  is  each  one's  part  of  the  gain  ? 

Ms.A'spart£65,  B's  £97  10s.  C's£l62  10s. 
5.  Three  partners,  A,  B,  and  C,  shipped  108  mules  for 
the  West-Indies  5  of  which  A  owned  48,  BS6,andC  24. 
But  in  stress  of  weather  tlie  mariners  were  obliged  to 
throw  45  of  them  overboard ;  1  demand  how  much  of  the 
loss  each  owner  must  sustain  ? 

JIns.  Jl  20,  i?  15,  and  0  10. 

4.  Four  men  traded  with  a  stock  of  800  dollars,  bj 
which  they  gained  307  dols,  A's  stock  was  140  dols.  B^ 
£60  dols.  C's  SOO  dols.  I  demand  D's  stock  and  what 
each  man  gained  by  trading  ? 

dns.  JJ^s  stock  was  glOO,  and  A  gained  g53,  72c/s.  5m. 
B  g99,  77icts,  CS115, 12^cis.  and  I)  gS8,  S7jicts, 

5.  A  bankrupt  is  indebted  to  A  21 1/,  to  B  300/.  and  to 
C  391/.  and  hitt  whole  estate  amounts  only  to  675L  10s. 
which  he  gives  up  to  these  creditors ;  liow  mucn  must 
each  have  in  proportion  to  his  debt  ? 

^ns.  Jl  must kaverioQ  C:>'.  3 Ja'.  B  £^£4  15s.  4-.<*  ana 
G  £2Q2  IG3  Sii 


140  COMPOUND    f  i:LLO,WSi£li». 

6.  A  captain,  mate  ana  20  seamen,  took  a  prize  worth 
5501  dols.  of  which  the  captain  takes  11  shares,  and  the 
mate  5  shares  ;  tho.  remainder  of  the  prize  is  equally  di- 
vided among  the  sail^irs ;  h©w  much  did  each  man  re- 
ceive ?  §5     cts, 

Ans.  Tiie  captain  received     1069,  75 
1-^ie  uiaic  486,  25 

Each  sailoi-  97,  25 

7.  Divide  the  number  of  360  into  3  parts,  which  shall 
be  to  each  other  as  2,  5,  and  4.    Arts,  80,  120.  and  160. 

8.  Two  mei  chants  iiave  gained  450/.  of  v/hich  A  is  t^ 
(.ave  3  tijues  as  much  as  15 :  hov»'  much  is  each  to  have  ? 

Ais.  J  £537  105.  and  B  £   il2  105.--^1+S=:4  : 
450  :  :  3  :  £337  iO-s.  A's  shci::e. 

9.  Three  persons  are  to  siiare  600/.  A  is  to  have  a  cer- 
tain sum,  B  as  much  again  as  A,  and  C  tliroe  limes  as 
mucli  as  15.     I  demand' each  man's  part  .^ 

Ans.  A  £66|,  B  £153f,  and  G  £400 

10.  A  and  B  traded  together  and  gained  100  (U)is.  A 
put  in  640  dols.  B  put  in  so  much  that  he  must  receive  GO 
dols.  of  ihe.  gain  ;  I  demand  B's  stock  ?        Am.  S960 

11.  A,  E,  and  C,  traded  in  company :  A  put  in  140  dols. 
B  250  dols.  and  C  put  in  120  yds.  of  cloth,  at  cash  price-;  • 
tliey  gained  230  dols.  of  which  C  t<5ok  100  dols.  for  hii 
share  of  the  gain  ;  hovv  did  C  value  his  cloth  per  yard  in  ■ 
I  ommon  stock,  and  wiiat  was  A  and  B's  part  of  the  gain  ? 

Ans,  C  jmi  In  the  cloth  at  S2^  per  ijard,    Agaiiicui 
S46,  67c/.'^'.  6;jt.4-  and  B  B83,  oZcts.  5?n,-r 

■  ■■!!■  ■!  I  ll■m^T^^l»1lm«l<■^«r^^»^¥T^lTa"^^-^•^^^^-»'-^»''^■~^'«n'^^""^»«^^^  i      ■  ■  iiiiii  i  ii  mm  niM 

COMPOUND   FELLOWSHIP, 

Or  Fellowship  with  time,  is  occasioned  by  scvoial 
shares  of  partners  being  continued  in  trade  an  une(|uai 
term  of  time. 

RULE. 
Multiply  each  man's  stock  or  share  by  the  time  it  v^aa 
continued  in  trade :  th.en. 

As  the  sum  of  the  several  products, 

Is  to  the  whole  gain  or  loss  : 

So  is  each  man's  particular  product, 

To  his  particular  share  of  tho  gain  or  loss. 


)i:ni)  i^KLLO'.vsnrp.  I  i- 


KXAMPIKS. 


£•    ^'?. 

d. 

;  3     3 

4  A'spt. 

;  6     6 

8  B-s  — 

:  9  10 

0  C's  — 

1.  A,  C  and  C  hold  a  pasture  m  common,  for  wliich 
Lliej  pay  19/.  per  annum.  A  ])ut  in  8  oxen  for  6  weeks  ; 
B  liZoxen  for  8  weeks;  and  C  12(|^cn  for  12  weeks ; 

what  must  each  pay  of  the  rent  ?        ^'^'''■ 

sx  6=  48")  r  h 

12x  8=  96  I    96  : 

'  \vlo=^l44  Las  288  :  19/.  :  :<;  144 

Sum  288J  tProofl9  0     0 

'Z,  Two  merchants  traded  in  company ;  A  put  in  215 
dols.  for  6  months,  and  B  390  dols.  for  9  months,  but  by 
misfortune  they  lose  200  dols.  ;  how  must  they  share  the 
loss?  Am\  d's  loss  S5S,  75cts.  B's  S146,  25cts. 

3.  Threft,  ]X'rsoR3  had  received  665  dols.  interest :  A 
had  put  in  4000  dols.  for  12  months,  B  SGOO  dols.  for  15 
months,  and  C  5000  dx)ls.4br  8  months  :  how  much  is  each 
man's  part  of  the  interevt  } 

Ms,  A  S240,  B  SS.25  and  C  S200  ^ 

4.  Two  partners  gained  by  trading  llOL  12s.  r  A's 
stock  was  120/.  lOs.  for  4  months,  and  B's  200/.  for  6i 
months;  "what  is  each  man's  part  of  tlie  gain  ? 

Ms.  A's  part  £29  18s.  S^^.^^Af.  B's  £80  lSs,Sid.,^^^^. 

5.  Two  merchants  enter  into  partnership  for  1 8  months. 
A  at  first  put  into  stock  500  dollars,  and  at  the  end  of  8 
months  he  put  in  100  dollars  more  ;  B  at  first  put  in  800 
dollars,  and  at  4  month's  end  took  out  200  dols.  At  the 
expiration  of  tlie  time  they  find  they  iiave  gained  700  dol- 
lars ;  what  is  each  man's  share  of  the  gain  ? 

.^     ^  ^^24,  07  4-J-.TS  share, 
''^^'  •    I  SSr5,  92  5-tB's.     do 

6.  A  an<l  B  companied ;  A  put  in  the  first  of  January, 
1000  dols.;  but  B  could  not  put  in  any  till  the  first  of 
May ;  what  did  he  tlien  put  in  to  have  an  equal  share 
with  A  at  the  year's  end  ? 

Mo,       g  Jib.  S 

As  12  :  1000  :  :  8  :  1000x12=1500  Jlns. 

.0 


The 


I4§  BOUBLii  Rur,r.  of  thref,, 

DOUBLE  RUI;E  of  THREE. 

Double  Rule   of  Tlircc  tcaclics  to   resolve   at 

once  such  questions  as  require  two  or  ::"fiore  statings  ia 
simple  proportion,  -wlietber  direct  or  inverse. 

In  this  rule  tliere  are  alv/iiys  five  terms  given  to  find  a 
sixtlij  the  three  firit  terms  of  which  arc  a  supposition, 
the  two  last  a  demaucL 

RULE. 

In  stating  the  question,  place  tlie  terms  of  the  supposi- 
tion so  that  the  principal  cause  of  loss,  gain  or  action  pos- 
sess the  first  place;  that  which  signifies  time,  distance  of 
place,  Ike.  in  the  second  place  5  and  i}\Q  remaining  tenii 
m  the  third  place.  Plac*5  the  terms  of  demand,  under 
those  of  the  same  kind  in  the  supposition,  if  the  blank 
place  or  term  bought,  tall  under  the  third  tcrift,  tl^e  pro- 
portion is  direct;  then  multiply  the  first  and  second 
terms  together  for  a  divisor,  and  the  other  iAivaa  for  tt 
dividend  :  but  if  the  blank  fall  under  the  first  or  second 
term,  the  proportion  is  inverse  ;  then  multiply  the  third 
and  fourth  terms  toother  for  a  divisor,  and  the  other 
three  for  a  dividend," and  the  quotient  will  be  the  answer, 

EXA  MPLES. 

L  If  7  men  can  build  S6  rods  of  wall  in  S  days;  liow 
many  rods  can  20  men  build  in  14  days  ? 

7  :    S  :  :  36  Terms  of  supposition. 
£0  :  14  Terms  of  deuuind. 

S6 

84 

42 

504 
20 


7x3==2il)  10080(480  rods  Ms. 
2.  If  lOO/.principal  will  gain  6^.  interest  in  12  months, 
what  will  400u  gain  in  7  m^ntlis  ? 

Principal  iOOl.  :  12mo.  ;  :  61.  Int. 

400    :     7  Jlns.  ML 


•J^Oi.'JOlKED     PROPORTION.  343 

5.  ir  lOOL  ulll  gain  6/.  a  year;  in  ^vhat  tiii-iC  %vil! 
400Lgain  H/.        /;.      mo.       £. 
100  :  la  :  :    6 
400  :        :  :  14        dns,  7  iivontlSe, 

4.  If  400Z.  gain  14^  in  7  mouths ;  wha|;  is  the  rate  per 
cent,  per  annum  ?        £.    mo.     Lit,' 

400  :  r  :  :  14 

100  :  12  Ms.  £6.^ 

5.  What  Principal  at  6/.  per  cent,  per  annum,  will  gain 
3 4f.  in  7  mouths  r        £,    mo.    .  Int. 

100  :  12  :  ":     6 

7  :  :  14        ^   Ms.  £400. 

6.  An  usurer  put  out  S6l.  to  receive  interest  for  the 
^ame  ;  and  when  it  had  continued  8  months,  he  received 
urincipul  and  interest,  88/.  17s.  4d.  5  1  demand  at  wliat 
rate  per  cent,  per  ann.  he  received  interest  r  Ans.  5  per  ct. 

7.  if  20  bushels  cf  wheat  arc  sulHcient  for  a  ikmilj  of 
8  persons  5  months,  how  much  will  be  sufficient  for  4  per- 
sons 12  months  ?  Ms.  24  bushels. 

8.  If  30  men  perfwrn  a  piece  of  work  in  20  days  :  how 
many  men  v.ili  accomplish  another  piece  cf  vvork  4  times 
a3  large  in  a  iifth  part  of  tUa  (hue  r 

no  :  20  :  :   I 

4  :  ;  4  Ms.  600. 

9.  If  the  carriage  of  5  cwi;.  3  qrs.  150  miles,  cost  24 
dollars  58  cents  ;  what  must  be  paid  for  the  carriage  of 
7  cvvt.  2  qrs.  25  lb.  64miles  at  the  same  rate  ? 

Jlns.  S14,  OBcis.  6?n.-|- 

10.  If  S  men  can  build  a  wall  20  [eei  long,  G  i^et  high 
and  4  feet  thick,  in  12  days  ;  in  what  time  will  24  men 
butld  one  200  icet  long,  8  tect  high,  and  6  feet  tiiick  ? 

8  :  12  :  :  20x6x4 


200x8x6     80  da^s^  Ms. 


CONJOINED   PROPORTION, 

Is  wher.  the  coins,  weiglits  or  measures  of  several  coun- 
tries are  compare^^  intlie^me  qyest»on  .  <jr  it  i;i  Joiaiii 


l5Q  C5.0NJ0I^'ED     mOPOKTlw- 

several  antecedents  have  to  their  consequents,  the  pro. 
portion  between  the  first  antecedent  ano  the  last  conse- 
quent is  discovered,  as  well  as  the  proportion  between 
the  others  in  their  several  respects. 

Note. —This  rule  may  generally  be  abridged  by  can- 
celling equal  quantities,  or  terms  that  happen  to  be  the 
same  m  both  columns :  and  it  may  be  proved  by  as  many 
statings  in  tlie  Single  Rule  of  Three,  as  the  nature  of  the 
question  may  require. 

CASE  1. 

When  it  is  required  to  llnd  how  many  of  tlic  ilvst  sort 
of  coin,  w^eight  or  measure,  mentioned  in  the  ([ucstion, 
are  equal  to  a  given  quantity  of  llie  last. 
RULE. 

Place  the  numbers  alternately,  beginning  at  tlie  left 
hand,  and  let  the  last  number  stand  on  the  left  hand  col 
umn;  then  multiply  the  left  hand  column  cor.tinually  for 
a  dividend,  and  tlie  right  hand  for  a  divisor,  and  the*<juo- 
tient  will  be  the  answer. 

EXAMPLES. 

1.  If  1001b.  English  make  95lb.  Flemisli,  and  191b. 
Flemish  25lb.  at  Bologna;  how^  many  pounds  Engllijh 
are  equal  to  oOlb.  at  Bologna  ? 

lb,  lb, 

100  Eng.=^5  Flemish. 
19  Fie.  =25  Eolo-na. 
50  Bologna.  '    Tlien  95x25=2375  the  divisor. 

95000  dividend,  and  2375)95000(40  JIns. 

2.  If  401b.  at  New-York,  make  48lb.  at  Antwerp,  and 
SOlb.  at  Antwerp,  make  36lb.  at  Leghorn ;  how  many 
lb.  at  New-York  are  equal  to  1441b.  at  Lcgliorn  ?    . 

Ms.  lOOlh. 
S.  If  70  braces  at  Venice  be  equal  to  75  braces  at  Leg- 
horn, and  7  braces  at  Leghorn  be  equal  to  4  American 
yards ;  how  many  braces  at  Venice  are  equal  to  64  Ame- 
rican yards  ?       "  .  Jhis.  104^-^ 
CASE  IL 
When  it  is  required  to  find  how  many  of  the  last  sort 
of  coin,  w'elght  or  meiisure,  mentioned  in  the  question 
ai'C  equal  to  a  given  quantity  of  the  first. 


rule: 

Place  the  numbers  aUeniately,  beginning  at  the  lefl 
hand,  and  let  the  last  number  stand  on  the  right  hand ; 
then  multiply  tlic  first  row  Tor  a  divisor,  and  tlie  second 
for  a  dividend. 

K:;AMrLEs. 

1.  If  24lb.  at  New-I^ond(m  make  COlb.  at  Amsterdam, 
and  50lb.  at  Amsterdam  60ib.  at  Paris :  liow  many  at 
Paris  are  efjual  to  40  at  New-London  ? 

Left,      Might. 
24  =  20         20  X  60  X  40  =  48000 

50  =  GO  --=  40  .Ins. 

40         24   X  50     tr=:  1)300 

2.  Jf  501b.  at  New-York  make  45  at  Amsterdam,  and 
BOlb.  at  Amsterdam  make  103  at  Dantzic ;  iiow  many  lb. 
at  Dantzic  are  ctiual  to  240  at  N.  York  r    .^us,  278-1^ 

3.  if  20  braces  at  Lcj:;liorn  be  equal  io  11  vares  at 
Lisbon,  and  40  vares  at  Lisbon  to  80  braces  at  Lucca ; 
how  many  bract^s  at  Lucca  are  equal  tw  100  braces  at 
Leghorn?  *'2ns.  110 


jJ  Y  this  rule  merchants  know  wliat  sran  of  money  ought 
to  be  received  in  one  country,  for  any  sum  of  different 
specie  paid  in  another,  according  to  tlie  given  course  cf 
exchange. 

To  reduce  the  monies  of  foreign  nations  to  that  of  the 
United  States,  vou  mav  consult  the  followiu'^ 

\  "FABLE: 

Showing  the  value  of  tlie  monies  of  account,  of  foreign 
nations,  estimated  in  Federal  Money.*     S  cts. 

Pound  Sterling  of  Great -15ri tain. 

Pound  Sterling  of  Ireland, 

Livre  of  France, 

Guilder  or  Florin  of  theU.  Netljcrlands, 

Mark  Banco  of  Hamburgh, 

Rix  DoUai-  of  Denmark, 

""haws  U.  S.  j! 


4  44 

4 

10 

0. 

,185 

0 

59 

0 

S3I. 

1 

0 

152  *ixcrkA^:GB, 

Rial  Plate  of  Spain,  0  10 

IVlilrea  of  Portugal,  1  24 

Tale  of  China,  1  48 

Pagoda  of  India,  1  94 

Rupee  of  Bengal,  0  55i 

L  OF  GREAT  BRITAIN. 

EXAMPLES. 

1.  In  45^.  10s.  sterling,  how  many  dollars  and  cents  ? 

A  pound  sterling  being3=444  cents, 
Therefore—As  IL  :  444cts.  ;  :  45,5/.  :  20202cfs.  dns. 

2.  In  500  dollars  how  many  pounds  sterling  ? 
A»444cts\  :  IL  :  :  50000cfs.  :  112^.  I2s.  3rf.-h  .9n.^. 

II.  OF  IRELAND. 

EXAMPLES. 

1.  In  901,  TOs.  6d.  Irish  money,  how  many  cents  ^ 
IL  Irish  =r410ci5. 
£.    cts.  £.  cts.  S     cts, 

Therefcft-e— As    1  :  410  :  :  90,525  :  S7115^=3f  1,  15i 
£•  In  168  dols.  10  cts.  how  many  pounds  Irish  ? 
As410cf5.  :  1/.  :  :  iCSlOcfs.  :  £41  Irish.    Jns. 

III.  OF  FRAiSCE. 

Accounts  are  kept  in  livres,  sols  and  deniers. 
512  deniers,  or  pence,  make  1  sol,  or  shiliiug, 
1 20  sols,  or  shillings,       —    1  livre,  or  pound. 

EXAMPLES. 

1.  In  250  livres,  8  sols,  how  many  dollars  and  cclits  ? 

1  livre  of  France=18T^-  cts.  or  185  mills. 
£•    '^'^'  £'  '^"-         S»  (^ts.  711. 

As  1  :  185  :  :  250,4  :  46524=46,  32   4    Ans, 

2,  Reduce  87  dols.  45  cts.  7  m.  into  livres  of  France^ 
mills,     liv,      mills,     liv.   so.  den. 

As  185  :  1  :  :  8745r  :  472  14  94-  Jlns. 

IV.  OF  THE  U.  NETHERLANDS. 

Accounts  are  kept  here  in  guilders,  stivers,  groats  and 
pfennings. 

r  8  phennings  make  1  groat. 
<    2  groats  —       1  stiver. 

(^20  stivers        —       1  guijder,  or  florin 
A  guL'der  is =^9  cent§.  or  390  mills. 


F.XAMri.F.^. 

iicJucc  19A  ;*iiiki'r;.s,  14  stivers,  into  federal  money. 
Guil,  cts.      GuiL      S    d-  c.  m. 
As  1    :   S9  :  :  124,7  :  48,    G   3    3     ^?«.<?. 
milh,     G.       vdJls.       G, 
As  S90  :  i  :  :  48633  :  124,7'  Froof. 

V.  OF  IIAMUUllGH,  IN  GERMANY 

Accounts  cire  kept  in  Hamburgh  in  marks,  sous  antl 
(leniers-lubs,  and  by  some  ia  rix  dollars. 

fl2  denicr's-lubs  r.iakc  1  sous-lubs. 

■i  \i)  so;?s-!i;h^.         —     1  niark-lubs. 

(^   3  ?\tark-hi'..s.       —      I  rix-doliar. 
NoTK.. — A  ii.wirk  Is  --  33-V  ct..^^.  oi*  just -?  of  a  dollar. 

RULE. 
Divide  Ihc  luaiksby  3,  the  quotient  will  be  dollars. 

KXAMl'LKS.  w. 


Reduce  G41  n:arks,  8  sous,  to  federal  raoney.       A> 

S)641,j  ;,.V 

S213,83r.  ^/7.c:. 
But  to  reduce  Federal  Money  into  Marks,  multiply 
the  given  sum  by  3,  &c. 

lleduce  121  dollars.  90  els.  into  marks  banco. 
i2l,90 


565,r0=365  ir.nrks  II  sous,  £,4  den.  Jiri^.  / 

VI.  OF  SPAIN. 

Accounts  are  kept  in  Spai  n  i rij^iastres,  rials  and  maiTadie«. 
^34  marva(ii?.\s  of  rihJj^  make  1  njal  of  plate. 
?.   Serials  of  plafe  —      1  piastre  oi  piece  of  8. 

To  reduce  rials  of  plate  to^Federal  Money. 
Since  a  rial  of  plate  is  =  10  centsj  or  1  dirae,yoiJ  need 
only  call  the  rials  so  many  dimes,  and  it  is  done. 

KXAMrLP.S. 

485  ria:snr:r485  i!iir.rs,^-48  do^s.  50  ci«;.  5&c. 


Sul  to  reduce  cents  into  rials  of  plate^  divide  1>>'  10-— 
llras,  345  cents-f-l 0=84,5  =84  rials,  17  maiTadieSj  fcc4' 

YII.  OF  PORTUGAL. 

Ax:counts  arc  kept  tliroughoiit  this  kingdom  in  milreas^ 
and  reas,  reckoning  1000  re;is  to  a  miirea. 

Note. — A  miirea  is  =  124  cents  ;  therefore,  tQ  reduce 
milreas  into  Fe<leral  Money,  multiply  by  124,  and  the 
product  will  be  cents,  and  decimals  of  a  cent. 

EXAMPLES. 

1.  In  r;:';  ^..\]]rc:i^  hnw  m:\nr  cents? 

C.  Ill  Q!  j  iinire'i^.   .... .....  liow  many  cents  ? 

Kci.:. — WliMi  the  reas  are  less  than  IGO,  place  a  cy- 
pher before  them.— Thus,  21 1,048  x  124 «=£6 169,952  cts, 
or  261  dols.  G9cenL->.  9  mills.  +  Ms, 

Bu^  to  re(iuce  cents  into  milreas,  divide  them  by  1£4'; 
and  if  decimals  arise,  you  must  carry  on  the  quotient  as 
far  as  lluee  decimal  places ;  then  the  whole  numbers 
thereof  will  he  the  milreas,  and  the  decimals  will  betlic 
reas. 

EXAMPLES. 

1.  In  4195  cents,  how  many  mili-eas  ? 

4 1 95 -i- 1 24  =33,850  -f  or  ^Smilr.  ^^Qreas.  Jins. 

2.  In  24  dols.  92  cts.  how  many  milreas  of  Portugal  ? 

Ans.  20  milreas^  096  reas. 

YIIT.  EAST  minx  ]\IOjir&Y. 

To  reduce  india^iwine^^  to  Federal,  viz. 

["Tales  of  Ciiiaa,  m;^ltiply  v/ith        148 
-^,  Pagodas  of  India,"  194 

(^  Rupee  of  Bengal^  55^ 

l<:XiOi?i'LES. 

1.  In  641  Tale s/m'Chjiia,'hcw  many  cents? 

'       £    ^  .  *'?»'^«  94863 

2.  In  50  Pagodas  ^^India,  how  many  cents? 

7.       '  '       Am.  9r00 

5.  In  98  liunces  of  Bengal,  lw)w  mauv  cents? 

^       Jim.  5439 


VULGAii  FRACTIONS. 

Having  briefly  introduced  Vulgar  Fractions  imme* 
diatcly  after  reduction  of  whole  numbers,  and  given  some 
general  definitions,  and  a  few  such  problems  therein  as 
were  nocessary  to  prepare  and  lead  the  scholar  immedi- 
ately to  decimals ;  the  learner  is  therefore  requested  to 
read  those  general  definitions  in  page  74. 

Vulgar  Fractions  are  either  proper,  improper,  single, 
compound,  or  mixed. 

1.  A  single,  simple,  or  proper  fraction,  is  when  the  nu- 
merator is  leas  than  the  denominator,  as  ^  :|  |  ||,  &c. 

C.  An  Improper  Fraction^  is  when  the  numerator  ex- 
ceeds the  denominator-  as  4  -J  V^,  &c. 

3.  A  Compound  Fraciion^  is  the  Traction  of  a  fraction, 
coupled  by  tlie  word  of,  thus,  |  of  -jI^j  ^  ^^  I  o^  l^  ^^' 

4.  A  Mured  A\miher^  is  composed  of  a  whole  number 
^id  a  fraction,  thus,  8 J,  14|^,  oic 

5.  Any  whole  number  may  be  expressed  like  a  fraction 
bv  drawing  a  line  under  it,  and  putting  1  for  denoniana- 
tor,  thus,  8=f,  and  IGthuj-:,  \S  &c. 

6.  The  comm(;n  measure  of  two  or  mere  numbers,  is 
that  number  v/liich  v»'iil  divide  cacli  of  thcin  without  a 
remainder;  thus,  3  is  the  common  measure  of  12,  24 and 
SO;  and  the  greatest  number  which  will  do  this,  is  called 
the  greatest  common  measure. 

7.  A  number,  winch  can  be  measured  by  two  or  more 
rmmbers,  is  called  their  common  midilple :  and  if  it  be  the 
least  number  that  can  be  so  measured,  it  is  called  the  least 
common  multiple:  thus,  24  is  the  common  multiple  of  2, 
S  and  4 ;  but  tlicir  least  common  multiple  is  12. 

To  find  iho  least  common  multiple  of  two  or  more 
r  limbers. 

RULK. 
..  D'.vi'Ie  by  any  number  that  wllldiviue  ^v/o  or  more 
CMC  given  numbers  without  a  remainder,  and  set  tli« 
«>tient::,  togetl;er  with  tLeui»'Jivided  rii;iiiber;>.  in  aline 
^:  nealh. 

2.  liividc  the  second  ll:t-e«  as  bc^fmv,  p.vaI  so  on  tiH 
il^^re  are  nwtwo  nuTiibei^hat  can  be  divided}  ^hen  the 


/ 


io6  JIEBUOTION   OF    VULGAR     FKAOTiONS. 

IHi  continued  product  of  the  divisors  and  quotients,  'wiligive 
^^  tlic  multiple  required. 

EXAMPLES. 

I.  *V\Tiat  is  tlic  least  common  multiple  of  4, 5, 6  and  10  ? 
Operation,         x5)4    5    6    10 

\  X2)4    16      2 

X2    1X3       I 

5x2x2X3=60  Jns. 
S.  What  is  the  least  common  multiple  of  6  and  S  ? 

Jns,  24 
5.  What  is  the  least  uumbci?  that  3,  5>  8  and  12  will 
measure?  ^fJns,  120 

4.  What  is  the  least  number  that  can  be  divided  by  the 
9  digits  separately,  v/ithout  a  remainder  ?    Aiis,  2/^20 

REDUCTION  OF  VUl.GAIi  LRAGTIONS, 

IS  the  bringing  them  out  of  one  form  iato  anothci',  in 
order  to  prepare  them  for  the  operation  of  Addition,  Sub- 
traction, &c. 

CASE  I. 

To  abbreviate  or  reduce  fractions  to  their  lowest  terms. 
RULE. 

1.  Find  a  common  measure,  by  dividing  the  greater 
tcvmby  the  less,  and  this  divisor  by  i:he  remainder,  and 
so  on,  aUvays  dividing  the  last  divisorjby  tlie  last  remain- 
der, till  nothing  remains^  tlie  last  divisor  is  the  common 
measure.* 

2.  l>i vide  both  of  the  terms  of  the  fraction  by  the  com- 
mon measure,  and  the  quotients  vvill  make  the  fraction 
required. 

*']Sojind  the  greatest  common  measure  of  viore  than 
two  numbers,  you  must  find  the  greatest  common  measure 
of  two  of  them  as  par  rule  ahovs.^  then,  of  that  common 
measure  and  one  of  the  other  numhevs^  and  so  on  through 
ell  the  numbers  to  the  lasts  then  wUHhe greSf^^t  co^n^ion 


1 


REJDUCTION   OF   VULGAR   TRAOTIONS.  157 

OR-^If  you  choose,  you  may  take  that  easy  method  in 
Problem  I.  (page  74.) 

EXAMPLES.  ^^ 

1.  Reduce  ^f  to  its  lowest  tenns. 
48)||(i  Operation. 

^l|4fl/6  common  mea.  8)f|=f  Ms. 


2.  Reduce  ^  to  its  lowest  terms.  Ms. 


^8 


Reduce  |f|  to  its  lowest  terms.  •fins.  ^ 

4.  Reduce  f  pfl  to  its  lowest  terms.  Ms.  i 

CASE  II. 

To  reduce  a  mixed  number  to  its  equivalent  improper 
fraction. 

RULE. 

Multiply  the  whole  number  by  the  denominator  of  the 
given  fraction,  and  to  t]\e  product  add  the  numerator, 
this  sum  v/ritten  above  the  denominator  will  form  the 
fraction  required, 

EXAMPLES. 

1.  Reduce  45}  to  its  equivalent  improper  fraction. 

45x8+r==3|7  Ms. 

2.  Reduce  19^  f  to  its  equivalent  improper  fraction. 

Ms.  V/ 

3.  Reduce  16^^^  to  an  improper  fraction. 

Ms.  x^^ 

4.  Reduce  6l-Jg  J-  to  its  equivalent  impropei  fraction. 

Ms.  2|«^5 

CASE  lU. 

To  faul  the  value  of  an  improper  fraction. 

Divide  ihQ  numerator  by  the  denominator,  and  the 
quotient  wiH  be  the  value  sought, 

EXAMPLES. 

1.  Find  the  value  of  V  5)48(9|  Jn$. 

2.  Find  the  v.ihie  of  W^  Ms.  IQif 
•^lEbjl  the  value  of  ^t  *^«5-  84^ 

i^gfS^^m]  tiie  value  of  %%^ *  ^ns.  6^^ 

\        '^  Find  the  value  of  V  •to^-  S 


..jS  reduction  of  vulgar  fhactions. 

CASE  IV. 

To  rciluce  a  wliole  number  to  an  equivalent  frd.ci'.gfly 
ing  a  given  denomiiiator.  * 

RULE. 

Multiply  tlie  whole  number  bv  tlie  given  den 
place  the  product  over  tlie  said  denominator,  i 
form  the  traction  required.  ff 

fc:XA:iPLES. 

1.  Reduce  7  to  a  fraction  wliose  deiuKiiinri'  o 
9.                           Thu-    '^ -^^-r63,  and  ^^  in- 

2.  Reduce  IS  to  a  vhose  denomi: 
be  12. 

3.  Reduce  100  to  Its  equivalent  fraction,  Ik  >/  ;    ^3 
(or  a  denominaJ:or.  '        Jln^.  ^ o.o  o  ^ 9 o o  ^,  v 

C  'VhE    V. 

To  reduce  a  compound  fraction  to  a  simple  one  r  ' 
vr.hie. 
KJ  -LE, 

1.  Reduce  all  whole  aiul  mixed  nuinbers.to  i~  " 
valent  fractions. 

2.  Multiply  all  (lie  numeiators  to^jether  (or 
merator,  vmd  ail  the  den<f,nJLiat'.MS  for  a  nev/ 
tor;  and  they  ^vill  form  the  fraction  recjiiircd. 

EXAM  PL  FS. 

1.  Reduce  -J  of  4  ^'^  ^  ^^  r?^  to  a  ^>iIT5<,'o  cnii- 

1x2x3x4 

■ ^^o^Tu    «^^5- 

2.  Reduce  4  of  |  cf  ^  Je  ^ctioa.     .' 

3.  Reduce  |  of  |^  (fi   ,3  .. .  a  single  fractir. 


v^ ;  I 


4.  Reduce  ioi'"  .     -  '  ,       ■     ^-  ^--^ion. 

5.  Reduce  ^  (-  v.i  a^:;.        iVaction. 


■)  2  6  G  0  . 
t.  u  0"     ' 


Note. — If  i\\e  denon.inator  u(  ai 
br)UGd  fractic!!  be  equal  to  tlu-  ^" 


ilJF.  D  \:  U  r  ! '.)  N     O  F     V  I.  \. I .  A  !<.     F  \\  A  <  "i  J.(. 


I^fl      ':•  thereof,  tliej  may  both  be  expuRgc.:,  www  tue 

r  ..embers  continually'  multiplied  (as  by  i}\t  rule) 

duce  t!ie  fraction  required  in  lower  t-erms. 

<m!ucc  %-  of  Of  4  to  a  simple  fraction. 

Tiius,  2x5 

4x7 
*  Malice  I  of  J  of  I  of  |4^  to  a  simple  fraction. 

CASK  VI.    : 

^'?  fractions  of  diifercnt  denominations  to  cr 
fiactiqi^s  having  a  ccinmondenominatoi. 


1.  lu'duce  alifraction.'^  to  simple  terms. 
•>.  Multiplj  each  numerator  into  all  the  denominators 
o}:t  its  ov.n,  for  a  nev/  numerator  ;  and  all  the  denom- 
,;oi.'^  into  each  other  continually  for  a  common  denom- 
.  U>i  ;  this  Avritten^'tuidei-  tlie  :'  iiew  Btuuerators, 

:  1  ^^ive  the  fractions  recpiired. 


EXAMPI 


^:duce  \\\  to    c; 
I  denominator. 


iractions,  having  a 
'e  nominator. 


16       IS  nev/  muncnitors 


24       lw4 
Reduce  ^ 


r-.  Reduce  i  2 


dem/nduat  «*s. 
and  W  to  a  comnuiJi  dfMiorninator. 

J?'iO  ■  »'-*<l     ^i^*    /IW//    8  80 
*^...'.    j,.^    5^0    '''^"     9^ 

f  andT^ioa  coiniDon  denominator. 
<»  .J'..?.  js|^  -^^-j  -j^-g-  ana  -5;,^^ 


M  REDUCTION    OF    VULGAR    FRACTIONS. 

4   Reduce  4  -^^  and  j\  to  a  coinmon  denominator. 
800      SCO  400  ^^ 

— ^'^'^ ;%  ro  ^^^^  A  ==^tV  •^W^- 

1000     1000         1000 

5.  Reduce  J  ^j  and  12-|  to  a  common  denominator. 

jiim     5  4    6  0    8  8  8 

.  fj<t.>.  Y2;  Ti    T'i 

6»  Reduce  |  -|  and  |  of  ||  to  a  ccniino:i  denominator. 

i]po        768      259219R0 

..^/..S.  -s-^jrsg-  ^--tTIT  T4T^ 

The  foregoing  13  a  general  Rule  for  reducing  fractions 
to  a  common  denominator;  but  as  it  will  save  much  la- 
bour to  keep  the  fractions  in  the  lov/est  terms  possible, 
tlie  following  Ride  is  much  preferable. 

For  reducing  fractions  to  t  -on  denominator. 

(By  Rule,  page  155)  find  tlie  least  common  multiple  of 
all  the  denominators  of  the  given  fractions,  and  it  will  be 
the  common  denominator  required,  in  which  divide  each 
particular  denominator,  and  multiply  the  quotient  by  its 
own  numerator  for  a  new  numerator,  and  the  new  nume- 
rators being  placed  over  the  common  denominator,  will 
express  the  fractions  required  in  their  lowest  terms. 

EXAMPLES. 

1.  Reduce  1 4  and  I  to  tl\eir  least  common  denomina- 


4)2 


£)r2     1     £ 

111     4x£=8  the  least  com.   denominator. 


8-7-£xl=4  the  1st.  numerator. 

8-r-4x3=6  the  Sd.  numerator. 

^  8-~8x5=5  the  3d.  numerator. 

These  numbers  placed  over  the  denominator,  give  the 
answerlff  equal  in  value,  and  in  much  lower  terms 
tlian  the  general  Rule,  whicli  would  produce  H  |^  |^ 
2.  Reduce  |  %  and  -^^  to  their  least  commoa  tienomi-* 


X' 


UEDUOTION  OF   VULOAR    FRACTIONS. 


5^  ^  -J  anu  -fj 
nomiuator.  *^ns.  ^  ,\  ^|  ^f 

4.  Reduce^  f  f  and  ^^^  to  their  least  common  denom 
inator.  Jlns.  ^\  fi  ^  ^-V 

^ASE  VII. 

To  reduce  the  fraction  of  one  denomination  to  tlie  fractio 
of  another,  retaining  tlie  same  value. 

RULE. 

Keduce  the  given  fraction  to  such  a  compDund  one,  as 
!i  express  the  value  of  tiie  given  fraction,  bj  comparing 
vith  all  the  denominations  between  it  and  that  denomi- 
ien  vou  would  reduce  it  to;  lastly,  reduce  this  com- 
ind  iVaction  to  a  single  one,         '        Y. 

EXAMPLi..-. 

^ .  Reduce  J  of  a  penny  to  the  fiaction  of  a  pound. 
By  comparing  it,  it  becomes  |-  of  ^\  of  r;\  of  a  pound. 

5  X     1   >f    1  "5     ^ 

i-.,4M^    =   ,^«,?, 

6  X-  1^^  20        1440 

2.  Keduce  j^tj  of  a  pound  to  the  fraction  of  a  penny. 

Compared" thus,  T-^  of  \o  of  Y'd. 
Then        5  x  20  x  Is 

440         1  I       '^"\y 

5.  Reduce  ^  ©fa  fartiiiii,^  to  th^  iraci;  on  of  a  shilling. 

*.  Ans.  ^,*;. 

4.  Reduce  |  of  a  shilling  to  tlie  fraction  of  a  pound. 

5.  Reduce  f  of  a  pwt,  to  the  fraction  of  a  pound  troy. 

6.  Reduce  f  of  a  pound  r^irdupois  to  the  fraction  ol 

7.  What  part  of  a  pound  avoirdupois  is  -j^i^  oF  a  cvrt.  ? 
Compounded  t\\v^.  r^,.  of  t  f >t  ^^  =-}-|-| = «  ^/J/j^?. 

S.  Wiiatpartof  ar  '  iM'k? 


iOB  REDUCTION  05   VULGAR  rRA0TiONS# 

9.  Reduce  i  of  a  pint  to  the  fraotioii  of  a  hbd. 


10.  Reduce  ^  of  a  pound  to  the  fraction  of  a  guinea 

Compounded  thus,  |  of  ^  of  tj^^^t  •^^^• 

11.  Express  5^  furlongs  in  the  fi^Gtion  of  a  mile. 

Thus,  5^^^^  of  1=11-  Ans. 

12.  Reduce  |  of  an  English  crown,  at  6s,  8d.  to  the 
fraction  of  a  guinea  at  28s.  dns.  A  of  a  p^idma. 

CASE  Yill. 

To  find  the  value  of  a  fraction  in  the  known  parts  of  the 
integer,  as  of  coin,  weight,  mcHsuie,  6cc. 

RULE. 

Multiply  the  numerator  by  the  parts  in  the  next  inferi- 
or denomination,  and  divide  the  product  by  the  denomina- 
tor ;  a^d  if  any  thing  remains,  multiply  it  by  the  next  in- 
ferior denomination,  and  divide  by  the  denominator/ ai 
before,  and  so  on  as  far  as  necessary,  and  the  quotfcnt 
will  be  the  answer. 

Note. — This  and  the  following  Case  are  the  same 
vrith  Problems  II.  and  III.  pages  75  and  7^ ;  but  for 
the  scholar's  exercise,  I  shall  give  a  few  more  examp^  s 
in  each. 

EXAMPLES. 

1*  What  is  the  value  of  |}^  of  a  poimd  ? 

Jns.  85.  9id. 
%  Find  the  value  of  J  of  a  cwt, 

II 

4.  How  much  is  ^Vt  ^^  ^  pound  avoirdupois  ? 

Ans.  7oz.  lOJr. 

5.  How  much  is  ^  of  a  hhd.  of  wine  ?  Ms^  45gals. 
6*  What  is  the  value  of  |f  of  a  dollar  ? 

jflfws.  55. 7id, 
7.  What  is  the  value  of  j^  of  a  guinea  ?  Jlns.  18s. 


Ms.  Sqrs.  Sib.  loz,  l^dr 
S.  Find  the  value  of  |  of  6s.  6(1.  Ms.  5s.  Oid. 


..iiDlTXOn   or   VULGAH   FIlACTIO^b,  .  153 

8.  Required  the  value  of  ^J^  of  a  pound  apotliecaries. 

^ns,  ^oz,  Sgrs. 

9.  How  much  is  ^  of  5L  9s.  ?       Ms.  £4  ISs.  5\d. 
10.  How  much  is  ^  of  §  of  J  of  a  hogshead  of  wine  ? 

0  dris.  ISgals.  Sqt&, 

CASE  IX. 

To  reduce  any  given  quantity  to  tlie  fraction  of  any  grea^ 

er  denomination  of  the  same  kind. 

[See  the  Rule  in  Problem  III.  page  75.'] 

EXAMPLES    FOR    EXERCISE. 

1    Reduce  12lb.  3oz.  to  the  fraction  of  a  cwt. 

2.  Reduce  IScwt.  Sqrs.  £Oib.  to  tlie  fraction  of  a  ton. 

Jlns.  11 

3.  Reduce  16s.  to  the  fraction  of  a  guinea,    •^ns.  ^ 

4.  Reduce  1  hhd.  49  gals,  of  wine  to  the  fraction  of  a 
fun.  Jlns,  ^- 

5.  What  part  of  4cwt.  Iqr.  24Ib.  is  3cwt.  Sqrs.  17lb. 
8oz.  Jlns.  I 


ADDITION  OF  VULGAR  FRACTIONS. 

RULE 

REDUCE  compound  fractions  to  singl^||es ;  mixed 
numbers  to  improper  fractions ;  and  all  oI|^^^  to  their 
least  common  denominator  (by  Case  YI.  KITle  II.)  then 
the  sum  of  the  numerators  written  over  the  common  de- 
nominator, will  be  the  stjm  of  the  fractions  required. 

EXAMPLES. 

1.  Add  5^  I  and  f  of  J  together. 

•^i-V  and  f  of  I .=-41 

Then  y  |  \^  reduced  to  theii-  l^^ast  coiunion  denominator 

by  Case  VI.  Rule  \\   \Jf!  bef:oiiie  Vr   il  i'^ 

Then  ir>!2+18+14«:i^y=6ff  or  (3|.  Jinswer. 


..ii  ADDITIOK    OF    VULGAR    FiiACTIONS; 

2,  Add  4 1-  and  |  togetlier.  Ms.  1| 

5.  Add  i  I-  and  f  together.  Ms.  1| 

4.  Add  1^  3§  and  41  together.  Ans.  '20\^ 

5.  Add  1  of  95  and  ■}  of  14i  together.  ^^ns.  44^ 

Note  1. — In  adding  mixed  nuinbers  that  are  not  com- 
pounded with  other  fractions,  you  may  first  find  the. sum 
of  the  fractions,  to  which  add  the  whole  numbers  of  the 
given  mixed  numbers. 

6.  Find  the  sum  of  5|  7|  and  Id] 

I  find  the  sum  of  |  and  |  to  be  |^=UJ- 

Then  I|^-f5-r^+15rnr28|j  Anf^. 

7.  Ajdd  I  and  ir|  together.  Ai:3.  17-^\ 

8.  Add  25,  8i  and  |  off  of -f^  Jlns.  SS-/^- 

NoTE  2.— To  add  fractions  of  moneys  weight,  &c.  re- 
duce fractions  of  different  integers  to  those  of  the  same. 

Or,  if  you  please  you  may  find  the  value  of  each  frac-  ^ 
lion  by  Case  VHI.  ii\  reduction,  and  then  add  them  in 
their  proper  terms. 

9.  Add  ^  of  a  shilling  to  |-  of  a  pound. 


2d  Method. 
|.£.=7s.  6d.  Oqrs. 
|s.  =0     6     ^ 

Ms.  8  0  ^ 

By  Case  VIIL  Reduction. 


1st  Method. 

^of^V==TlF:C- 

Then^|^+|.=:,'^//o£- 

Whole  value  by  Case  VIIL 

is  8s.  Od,  Sfqrs.  Ms. 

ID.  Add  IBr  Troy,  to  |  of  a  pwt. 

;  Ms.  7qz.  4pwi.  IS^-gr 

11.  Add  ^  of  a  ten,  to  -Z^-  of  a  cwt. 

Ms.  ISlcwt  \qr.  Uh.  VZ^^oz. 

12.  Add  A  of  a  mile  to  ^^^  of  a  furlong. 

.61ns.  Gfur.  ^Spo. 

IS.  Add  "  of  a  ynrd,  |  of  a  foot,  and  J  of  a  mrle  to- 
gether.        ''  ^  Ms.  1540yds.  ^ft  9in. 

14.  Ad«l  I  of  a  week,  ^  of  a/day,  |  of  an  hour,  and  -J-  of 
»  minmte  together.  Am^^a.  2ho.  SOmin.  iSsec 


sun:  OF    VULGAR     FRACTIONS. 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

RULE.*  ^ 

PllEPARK  the  fractions  as  in  Addition,  and  the  dif- 
ference of  the  numerators  written  above  tlie  common  de- 
i^ouiiuatorj  will  give  tht^difterenceof  the  fraction  requii-ed. 

EXAMPLES. 

I.  JVom  i  take  I  of  I 

I  of  |-=H=i^  ^^^^«n  '  and  J^^:^!^  ^ 

Therefore  9— r=-j%=-|-  ^^^^  *^^^' 

\  From  IJ  take  ^  Answers,      \\ 

:.  From  11-  take  y,  ^^ 

.  From  14  take  \}  13^% 

.  Whatift  the  diifcrei-ee  of  -,^^  arul  ]I  r  ^fy 

..  AVhatditTers  -j>^  fioui  ^  ?      '  -^V^ 

r.  From  14i  take  f  of  19  1^ 

8.  From  ||  take  -||^  0  remains, 

'j.  From  14  of  ammnd,  take  ^  of  a  shilling. 

5  of  ^V=Tlo^^•  Then  from  f|£.  take^-J^£.  Arts.  |}£. 

Note. — In  tractions  of  money,  weight,  occ.  you  may,  if 
you  please,  find  tiie  value  of  tlie  given  fractions  (by  Case 
VIIL  in  Reduction)  and  then  subtract  them  in  their  pro- 
per terms. 

10.  From -jV£- take  ^  sliilling.  Jlns.  5s.  6d,  2^qrs. 

II.  From  -f  of  an  oz.  take  J  of  a  pwt. 

*'9ns.  llpivt  3gr. 

12,  From  }  of  a  c>vt.  ta^ie  y^  of  a  lb. 

Jins.  Iqr.  mb.  6oz,  10^%dr. 

13.  Froju  ^  weekis,  take  f  of  a  day,  and  ^  of  f  of  ^  of 
an  hour.  •dns.  Sick  4da.  l2ho,  19 hum.  IT^sec, 

*In  subtracting  mixed  numbers,  when  the  lower  fraction  is 
greater  than  the  upper  one,  you  may,  without  reducing  them 
to  improper  fractions,  subtract  the  numerator  of  the  lower 
fraction  from  the  common  denominator,  and  tQ  that  difference 
add  th«  upjjer  numerator,  carrying  one  to  the  unit's  place  of 
t]<i  lowfcr  whole  number. 

llso,  a  fraction  may  be  snbtracted  from  a  wholo  number 
bj  taking  the  numerator  of  the  fraction  from  its  denomina- 
tor, and  placing  the  remainder  over  tlie  denominator,  then 
thkinpf  one  from  thfc  whoJe  number. 


\ 


)   MUI/n PLICATION  07  >  .    !  FRACTION^,. 


REDUCE  vr  Moers  to  the  improper 

fractions,   mixc  .}le  ones,  and  timseof 

dilTerent  integers  to  the  same  ;  tly^n  multiply  all  the  au- 
meratots  together  for  a  Fiew  numerator,  and  all  the  de- 
nominators together  foi-  anew  denorainator. 


Mr:[liply  f  b^  4 


£.  Muitlpij  |.  by  ? 
S,  .^liiltipiy  5i  by  i- 

4.  Multiply  I  of  7  bv  '* 

5.  Multiply  1^1  bv  >.  ■■ 

6.  Multiply  I'of  '  :'^ 
r.  Multiply  f!2  by 

Multiply  I  Gf^|  by  |  of  SI 


--'If 

23 
Y4 


9.  Wluitis  thQ  continued  product  of  J  of  !•  7,  5t^  and 

DIVISION  OF  VULGAR  FRACTIONS. 

RULE. 

PREPARE  iLe  fractions  as  before ;  then,  invert  iiie 
divisor  and  proceed  exactly  as  in  multiplication : — ^The 
products  will  be  the  quotijent  required. 

r,X  A  TRIPLES. 

4X5 

1.  Divi^^.'  I  by  J  Thus, =|^  diis. 

S  X  7 

2.  Divide  .]^  oy  v  dnsivers.  1-^^ 

5.  Divide  I  of  I  by  i  $ 
4.  What  is  the  quotient  of  17  in-  f  ?  *  59^ 
D.  Divide  5  by  fy  '    '                       7-J- 

6.  Divide  i  o'f  f  of  -J  by  }  of  i  3 J 

7.  Divide  4|-  by  f  of  4  ^ 

8.  Divide  71  by  127  ^V 
^   Divide  d205^  bv  -f  of  91  714 


KIJLE    OF    THUEF    DlilEC  I",   IKVEXlSE,   &C.  16T 

RULE  OF  THREE  DIRECT  IN  VULGAR 
FRACTIONS. 

RULE. 

i'REPAIU''.  tlie  iVaclions  as  before,  then  state  your 
question  agreeable  to  the  Rules  already  laid  down  in  the 
Rule  of  Three  in  whole  numbers,  and  invert  the  first  term 
in  the  proportion ;  then  multiply  all  the  three  terms  con- 
tinually together,  and  the  product  will  be  the  answer,  in 
the  same  name  with  the  second  or  middle  term. 

EXAMI'LES. 

1.  If  f  of  a  yard  cost  J  of  a  pound,  what  will  r^^  of  an 
Ell  English  cost? 

|yd.=A  of  ^  of  i^U  or  i  Ell  English. 
Ell.  £.     Ell.  s,  d.  qrs. 

\  2.  If  4  of  a  yard  cost  J^  ot  a  pound^what  will  4C^J-  yds. 
come  to?  .Tns.  £59  Ss.  6^d. 

3.  If  50  bushels  of  wheat  cost  17 f  I.  wliat  is  it  per  bush- 
el ?  Jtns.  7s.  Gd.  Hfqrs. 

4.  If  a  pistarcen  be  worth  14|  pence,  what  are  100  pis- 
"tareens  worth  ?  ^ns.  £6 

5.  A  merchant  sold  5^  pieces  of  cloth,  each  containing 
244  yds.  at  9s.  -^d.  per  yard ;  what  did  the  whole  amount 
to.^  '  Jlfii.  £60  lOs.  Od.  3|</rs. 

6.  A  person  having  |.  of  a  vesical,  sells  f  of  his  share  for 
:  :S12?. ;  wliat  is  the  whole  vei-sel  worth  ?        dns.  £780 
>.  7.  if  I  of  a  sidp  be  worth  f  of  her  cargo,  valued  at 
! 'I^OO^L  ^vhat  is  the  whole  ship  and  cai-go  worth  ? 

Arts.  £10031  14s.  lU\-fZ. 


iN  VERSE  PROPORTION. 

RULE. 

PREPARE  tlie  fractions  and  state  the  question  as  be- 
fore, then  invert  t!ie  tliird  term,  and  multiply  all  the  thre^ 
.    ns  togetlicr,  the  product  will  }Mi  the  answer^ 


.Lii  Oif 


1.  How  nuich  h)  "  h  J  van!  vih  \ 

yards  ot  cloth  whi( ;  ^  j.wd  wide  ? 

l'{/s.  ?/r/s.  yds.  ids. 

As    1|:'ai:;|  AndJxVx!-vW^l^A'^«^• 
2.  If  a  man  perform  a  journey  in  S-}  days,  v/hen  t'^e 
day  is  t2-|  hours  long ;  in  how  many  days  will  lie  do  \t 
^vhen  the  day  is  but  9ihoiirs.  '     Ans»^^  rrs-  daijs. 

S.  If  IS  men  in  11  |daj3,  rnow  ?:1 ,;  acres,  m  how  many 
days  will  8  men  do  i\\^.  sarnie.  ./^.s.  18||  day^. 

4.  How  much  in  leiigth  that  i^  ''  ^ -■  <:s  b-oad,  wii! 
make  a  square  foot  ?  r.  9.^1  wches, 

5.  If  25|.s.  ^-iU  ^^-^^  n>-  ^';e  Cu..:,,^.,  ...  un  cwt.  \A5\ 
miles;  how  f.:.  be  carried  for  the  saine  mo- 
ney ?  •    •.  "^'l^^  miles, 

6.  How  many  yards  of  baize  v  !  yards  wide, 
will  line  18J  yVnas  of  cainbU^^    -^ 


liVLl^  Oi^  THREE  DlRKv,  TALS 

RULE. 

REDUCE  your  fractions  to  decimals,  and  state  yciir 
question  as  in  wh./le  numbers  s  iKiultipiv  the  second  and 
third  to;;ether;  divide  by  the  iirsc,  aiid  liie  quotiont  ^vill 
be  the  answer,  &c» 


t.  If  I  of  a  vil. 
come  to  ? 


Gosfc  -j'^2  ^^3  pound  ;  what  will  15^  yds* 
i^,S75  y7^=,58S+and  ^-=5fj 
Yds.         £  14/5.      /;.  £.  s.  iL  ^r.5. 

As  ,875   :    ,583    : ,:    15,75  :  i0,494==10  9  10  2,e4Au3 
2.  If  1  pint  of  wine  cost  S,2s.  ^^'*'■■^  -"=r  13.5  hlids.  ? 

?.  If4ijyd3.  <:osii3s.4jd.  what  ;;.L  -^^   ' -^*  ' 


5 
SIMPLE   INTEREST   BY   DECIMALS.  li 

4.  If  1,4  cwt.  of  sugar  cost  lOdols.  9  cts.what  wUl  9 
cwt.  S  qrs.  cost  at  the  same  rate  ? 

civt,        S  cwt,     8 

As  1,4  :  :  10,09  ;  :  9,75  :  70,£69=g70,  9.6cts.  9?n.-f 

5.  If  19  yards  cost  25,75  dols.  what  will  435^  yards 
come  to  ?  diis.  |g590,  Qlcts.  T-^^m. 

G.  If  345  yards  of  tape  cost  5  dols.  17  cents,  5m.  what 
%vill  1  yard  cost?  Ans,  ,015n=licfs. 

7.  Ka  man  lays  out  121  dols.  23  cts.  in  merchandize, 
and  thereby  gmns  59,51  dols.  how  much  will  he  gam  in 
living  out  \  2  dollars  at  the  same  rate  ? 

..3?fS.  3,91  c/o/s.=S3,  91cfs.    ^ 

8.  How  many  yards  of  ribbon  can  I  buy  for  25^  dols. 
if  )Z^l  yds.  cost  4^  dollai-s  }  .Ins,  ITSh yards, 

9.  li  17Sh  yds.  cost  25i  dollars,  what  cost  5-9|  yards  ? 

Jlns,  mi 

10.  If  1,6  cwt.  of  sugar  cost  12  do^s.  12  cts.  what  cost 
S  hhds.  each  11  cwt.  5  qrs.   10,12  lb.  .= 

Jus.  269,0:2  t/o/s.:^S269,  7cis.  2m.+ 


SIMPLE  INTEREST 
A  TABLE  OF 

BY  DECIMALS. 

RATIOS. 

Hate  per  cent.  | 

Ratio,      1  Rale 

percent.  \ 

Ratio, 

I   i 

f            5       ^ 

.03 

,045        j 

,05          i 

?   1 

.055 

,06 

,065 

,07 

Ratio  is  the  simple  interest  of  1/.  for  one  year;  or  in 
fg^eral  money,  of  SI  for  one  year,  at  tht  rate  per  cent. 


agreed  on 


RULE. 


Multiply  the  Principal,  Ratio  and  iimo  continually  to- 
gethcr,  and  the  lasi  product  will  be  the  interest  required. 


ESASIPLHS. 


i.  Required  the  intcresiof  21 1  dob.  45  eta,  foi'  5  ywr* 
^  5  Ml  cent  mi  &nnmn  ? 


SIMPLE  INTEREST  BY  pECtMAJLS. 

S  cts, 
211 ,45  Principal. 
5O5  Ratio. 


lOjorSo  Interest  ior  (me  year, 
5  Multiply  by  the  time. 


52.8625  £ns.r=.^5Q,  SGcts.  ^m. 

2.  What  is  the  interest  of  645/.  iOs.  for  3  years,  at  6 
per  cent,  per  annum  ? 

/;64555x06x5=ll6,l90r:=/;ilG  Ss.  QtZ.  2,4(/rs.  .to. 

S.  What  is  the  interest  of  1.21/.  8s.  6d.  for  4^  years,  at 
J  per  cent,  per  annum  .^        ^^jis,  £32  ids.  8d.  lS6qr&. 

4.  What  is  the  amount  of  536  dollars  39  cents,  for  IJ 
yearSj  at  6  per  cent,  per  annum  ?  Ans,  S5 84,6651 

5.  Required  the  amount  of  648  dols.  50  cts.  for  ISJyrgi 
at  5^  per  cent,  per  annum  ?  Ans,  §1103,  26cfs,+ 

CASE   II. 

The  amount,  time  and  ratio  given,  to  find  tlie  principal, 

RULE. 

Multiply  the  ratio  by  the  time,  add  unity  to  tlie  product 

for  a  divisor,  by  which  sum  divide  the  amount,  and  the 

quotient  will  be  the  principal. 

EXAMFLES. 

1.  What  principal  will  amount  to  12S5j9r5  dollars,  in 
5  vears,  at  6  per  cent,  per  annum  .^     S  S 

,06x5-1-1=1,30)1235,575(950,75  Jns. 

2.  What  principal  will  amount  to  873/.  19s.  inQyeai'S, 
at  6  per  cent,  per  annum  ?  Arts^  £567  10?. 

3.  W^hat  principal  will  amoiiiitto  626  doi*6  cts.  in  12 
years,  at  7  per  cent.  ?       Ans.  S340,25=S340,  25ci5. 

4.  "Wliat  principal  will  amount  to  956/.  10s.  4,1 2od. 
in  8|  years,  at  5^  per  cent.  ?  Jhis.  £645  15s. *^ 

,  »       CASE  HI. 
The  amount,  principal  ami  liuie  given,  tcfind  the  ratio. 
RULE. — Subtract  tlie  principal  from  vhe  amount,  di« 
Tide  the  remainder  by  tlic  product  of  the  time  and  princi*' 
T^-^l,  and  the  quotient  will  be  the  ratio. 

EXAMPLES. 

-.  At  what  rate  per  cent,  will  950^75  doljf-aQiLOiinf  id 

■■  :^75  doI&  m  5  j'ears  ? 


Fi-om  the  amount    =    l!:35,9r5       ^  ^  "^ 
Take  the  principal  =      950,75 


950,75x5=47'55575)28552250f506=6perccnt 
.eS5,S250  Ans. 

2.  At  what  rate  per  cent,  will  567/.  10s.  amount  to 
875/.  19s.  in  9  years  ?  dns,  6  per  cent. 

3.  At  what  r«ite  per  cent,  will  340  dols.  25  cts.  amount 
to  6;J6  dols.  6  cts.  in  i:'2  years  }  Jns,  7  per  cent. 

4.  At  what  rate  per  cent,  will  645/.  15s.  amount  to 
D561.  10s.  4A9.5d,  in  8^^  years  t  Jlns.  5-^  2Je\'  cent. 

CASE  IV. 

Tiie  amoiuit,  f>iiiu:;pal5  and  rate  per  cj;iL  r:*'- cn  to  find 

the  time. 

RULE. 

Subtract  the  principal  from  the  amount;  divide  the 

remainder  bj  the  protiuct  of  the  ratio  and  principal :  and 

the  quotient  will  be  the  time. 

EXAMPLES. 

1.  In  wiiat  time  will  950  dols.  75  cts.  amount  to  1235 
'vollarsj  97,5  cents,  at  G  per  cent,  per  annum  ? 
From  the  amount     SI 235,975 
Take  the  principal      950,75 

«         ■ .,.,    ,  ■ 

950,75x06=:57.0450';285,2250(5  years,  M?, 

'  235,2250 


2.  In  what  time  will  567/.  lOs.  amount  to  875/.  19s. 
at  6  per  cent,  per  annum  ^  Jinn,  9  years. 

3.  In  what  time  will  340  dols.  25  cts.  amount  to  6^ 
dols.  6  cents  at  7  per  cent,  per  annum  ?  Ans.  \9^  years. 

4.  In  what  time  will  645/.  15s.  amount  to  956/.  10s, 
4.125d.  at  oh  per  ct.  per  annum  .^  ./^  -  'V*".-   v^""  -  -•— . 

•  -- — ^-^ — 

TO  CALCULATE  INTERE>ST  FOR  DAYS. 

;>  Multiply  tlie  principal  by  the  given  number  of  days, 
'  and  that  product  by  the  ratio  5  divide  tlie  last  product  W 
\  365  (the  number  of  days  in  a  year)  and  it  will  give  the 
\  injtere^trefiiiirc^ 


fl* 


SlMi-i^K   INTEREST  BY  DECTMALS.    ^ 
j  EXAMPLES. 

.  {What  is  the  interest  of  S60Z.  10s.  for  146  days^  at  6 
cent  ? 
S60,5xi46x,06      £,      £.  s.  d.  qrs, 

=8653= 


SG5 


565Sr=:8  13  0  1,9  Ms, 


2. 

at  6 

S. 

5pci 

4. 
days 


What  is  the  interest  of  640  dels.  60  cts.  for  100  dayi 
fer  cent,  per  annum  ^  Jlnfi.  310,530^5.+ 

Required  the  interest  of  250?-.  iTs.  for  120  days  at 
-  cent,  per  annum  ?      Ms.  £4,1235=4L  2s.  S^d.-i-  ^ 

Required  the  interest  of  481  dollars  75  cents,  for  25 

n'  '"  '      "r  annum  ?     Ms.  S-?  SOcfs.  9m,+ 


Is. 

^      '^ 

?2 

g< 

p. 

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o 

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1 

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o 

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1 

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CO 

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^* 

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00 

OO 

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00 

en 

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CJi     00 

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2" 

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8 

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CO       CO 
CO      O 

07          ,'ii 

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00 

l# 

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f^ 

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r>: 

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t^ 

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K> 

a 

siMr:/i:  .xTr.iiF.sr  ry  decimals. 


5 


W^'hcn  interest  is  to  be  calculated  on  casU  accounts,  &-c. 
^v]1ele  partial  pavments  are  made  :  maltiplj  ihfi  several 
balar.ccs  into  tiie  days  they  are  at  interest,  tlicn  multiply 
the  Sinn  of  these  products  by  the  rate  on  the  dollar,  and 
divide  the  ia^t  product  by  365,  and  you  will  have  the 
whole  interest  due  on  the  account,  &c. 

EXAMPLES. 

Lent  Peter  Trusty,  per  bill  on  demand,  dated  1st  ot 
Tunc,   1800,  ^000  dollars,  of  which  I  received  back  the 

^ih  oC  August,  400  dollars:  on  the  15th  of  October,  600 
liars  :' on  the  11th  of  December,  400  dollars ;  on  the 

.  rh  of  February,  1801,  200  dollars;  and  on  the  1st  of 
-.  i!  0,  ^00  doHars:  howniUi:h,interest  is  due  on  the  bill, 


rccK<inni^  at  b  i^.^v  cer 
i  BOO, 

June  1,  Principal  per  bill, 
August  19,  Keccived  in  lart, 


dniis,  da^js.  products. 


October  15. 


Balance, 
Ueceiveil  ia  part, 


2000 
400 

1600 
600 


Dec^i 


inbcr  1 1 


T^alance,     10 


57 


158000 


91200 


57000 


1801, 
F.cbruarv  1 


lune  J,  Rec'<i 


•e-l 


m  j)^ 


200 


Balance,    400 


68  !    40800 


104       41600 


-.88600 


Then  G8S600 

,06  Ratio. 


S  cfs.  m. 

S65V:^S3l6,00(o3,8r9  ^?z.s.  ==  05    87    9  -f- 

27ie  fallowing liule  for  computing  interest  on  any  nrde^ 

OT  ohllgation^  when  there  are  payments  inpartn  orend(/rse» 

ments^  teas  established  by  the  Superior  CGiirt  of  the  State 

of  Connecticut,  in  17S-1. 

RULE. 
*•'  CQinpute  tlie  interest  to  the  time  of  the  first  pay- 


174  SIMPLE   liVTKHEST   BY   DECI>rAt.^.  « 

ment;  if  that  be  one  year  or  more  from  the  time  the  in- 
terest commenced,  add  it  to  the  principal,  and  deduct  the 
payment  frotn  the  sum  total.  If  there  be  after  payments 
made,  compute  i\ie  interest  on  the  balance  due  to  the 
next  payment,  and  then  deduct  the  payment  as  above; 
ap.d  in  like  man-ier  froia  wae  payinent  to  another,  till  all 
the  payments  ai'^  absorbed  ?  provided  the  time  between 
one  payment  and  anotiier  be  one  year  or  more.  But  if 
any  payment  be  made  before  j)nt;  year's  intei-est  liath  ac 
crued,  then  compufe  th^  interest  on  the  principal  sum 
due  on  the  r»' :'  ar,  add  it  to  the  principal, 

and  compiitv  -  sam  paid,  from  the  time 

it^vaspaid,  up  to  Uia  .nd  ;.ttlie  year;  add  it  to  the  sum 
paid,  and  deduct  tiiatsui!!  IVointhe  principal  and  interest 
added  as  above, "^ 

"  If  any  pajuienis  he  rnadc  of  a  less  sum  than  thein^ 
terest  arisen  at  (he  tinie  of  such  payment,  no  interest  is 
to  be  computed  but  only  on  the*  principal  sum  for  anj 
period . ' '  "       Klrbij's  liiyaris^  page  49. 

E::A:.irLi:s. 
A  bond,  or  note,  dated  January  4th,  1797,  was  given 
for  1000  dollar.-,  iiit'orest  at  6  per  cent,  and  tliere  were 
payments  endorsed  up<;n  it  as  tollows,  viz.  §S 

1st  payment  FeUruarv  19,  i798.      '  200 

2d  payment  Jr,  ne  29,"  1 799.  500 

Sd  payment  November  14,  1799.  260 

I  demand  1k3w  much  remains  due  on  said  note  the  24th 
of  D^.cember,  1 800  ? 

1000,00  dated  .lanuary  -1,  1797. 
67,50  interest  to  February  19,  1798~13i  mo7?f/js. 

1067,50  amount.  [Carried  up 

*If  a  year  does  not  extend  beyond  the  time  of  final  settle- 
ment ;  but  if  it  does,  tiien  find  th'o  amount  of  the  principal  sura 
due  on  the  obligation,  up  to  the  tsni^  of  settlement,  aud  bkewiee 
find  the  amount  of  the  sum  paid,  from  tbe  time  it  was  paid,  up 
to  the  time  of  final  i^ettlement,  and  deduct  this  amount  from 
the  amount  of  the  principal.  But  if  tbere  he  several  payments 
made  witbjn  tbe  said  tinje,  find  tlie  amount  of  the  several  pay- 
ments, from  the  time  they  were  paid,  to  the  time  of  settlemenj^ 
wad  deduct  their  amount  from  the  amount  of  the  principal. 


SIMPLE   INTEREST  BY   DECIMALS.  175 

1067,50    amount.  [Brought  up, 

200,00    first  payment  deducted. 

867,50    balance  due,  February  19,  1798. 
70,845  Interest  to  June  29,  1799=16^  months. 


938,345  amount. 

^►00,000  second  payment  deducted. 

438,345  balance  due,  June  29,  1799. 
26-jj^^    Interest  for  one  year. 

464,645  amount  for  one  year. 

269-750  amount  of  third  payment  for  7^  months,* 

194,895  balance  due  June  29,  1800.       wo.  da, 
5,687  Interest  to  December  24,  1800,  5    25 


200,579  balance  due  on  the  Note,  Dec.  24,  1800. 
RULE  II. 
Established  by  the  Courts  of  Law  in  Massachusetts  for 
computing  interest  on  notes^  c^'c.  on  which  partial  pay" 
merits  have  been  endorsed, 

"  Compute  the  interest  on  the  principal  sum,  from  the 
time  when  the  interest  commenced  to  the  first  time  when 
a  payment  was  made,  which  exceeds  either  alone  or  in 
conjunction  witii  the  preceding  payment  (itany)  the  in- 
terest at  that  time  due  :  add  that  interest  to  the  princi- 
pal, and  from  the  sum  subtract  the  payment  made  at  that 
time,  together  with  the  precedin;^  payment  (if  any)  and 
the  remainder  forms  a  nesv  principal ;  on  which  compute 
and  subtract  the  payments  as  upon  the  lirst  piincipal, 
and  proceed  in  this  manner  to  the  time  of  final  settle- 
ment." 


8  cts. 
*260,00  third  payment  with  its  interest  from  the  time  it 
9j75        ivas  paid^  np  to  the  end  of  the  year^  or  fror 
'    ■■  ■•        JV  or.  I  i-   ;^99,  to  June  29, 1800,  which  i$  '     / 
£69,75  amount.  '^wi^ki       ! 


»r6  SIMPLE   INTEREST    BY   SECI^IALS. 

Let  the  foregoing  example  be  solved  by  tiiis  Rule. 

A  note  for  1000  dols.  dated  Jan.  4,  1797,  at  6  per  cent* 
1st  payment  February  19,  1798.  g200 

2d  payment  June  29,  1799.  500 

3d  payment  November  14,  1799.  260 

How  much  remains  due  on  said  note  the  24t]i  of  De 

cember,  1 800  ?  g  cts. 

Principal,  January  4,  1797,  1000,00 

Interest  to  Feb.  19, 1798,  (15^  Vdo.)  67,50 

Amount,     1067,50 
Paid  February  19,  1798,  200,00 


Remainder  for  a  new  principal,  867,50 

Interest  to  June  29,  1799,  (16^-  mo,)  70,84 

Amount,  938,34 

Paid  June  29,  1799,  500,00 


Remains  for  anew  principal,  438,34 

Interest  to  November  14,  1799,  (4i  mo,)  9,86 

Amount,  448,20 

November  14,  1799,  paid  260,00 

Remains  a  new  principal,               -  188,20 

Interest  to  December  24,  1800,  (15i  vio  )  12,70 

Balance  due  on  said  note,  Dec.  24,  1800,  200,90 

S  cts. 

The  balance  by  Rule  T.     200,579 

EyRule  11.     200,990 

Difterence,     0,4  U 

Another  Example  in  Rule  II. 

A  bond  or  note,  dated  February  1,  1800,  was  given  fo" 

500  dollars,  interest  at  6  per  cent,  and  there  were  pay 

ments  endorsed  upon  it  as  follows,  viz.  S  cts, 

1st  payment  May  l,  1800,  40,00 

2d  payment  November  14, 1800,  8,00 


COMrOUND    INTEREST   BY    DEOIMA1.S.  177 

Sd  payment  April  1,  1801.  12,00 

4th  payment  May  1,  1801.  v^0,00 

How  much  remains  due  on  sa'*!  »j«»m»  tJ^e  i-r..  o'    »«0(> 

tember,  1801  ?  6  •*». 

Principal  dated  February  1,  180U,  d'JO,00 

Interest  to  May  1,  1800,  (3  wo.)  7,50 

Amount,     507,50 
Paid  May  1,  1800,  a  sum  exceeding  the  interest,  40,00 

467,50 
year.)     _  28,05 


New  principal,  May  1,  1800, 
Interest  to  May  1,  1801,  (1  ; 


Amount,     495,55 
Paid  Nov.  4,  1800,    a  sura  less  than  the 

interest  then  due,  8,00 

Paid  April  1,  1801,       do.        do.       12,00 
Paid  Mav  1,  1801,  a  sum  greater,      30,00 

50,00 

New  principal  May  1,  1801,  445,55 

Interest  to  Sep.  16,  1801,  (4i  mo,)  10,02 

Balance  due  on  the  note,  Sept.  16,  1801.     ^455,57 
ICJ*  The  payments  beirtg  applied  according  to  thh  Rule, 
Jceep  down  the  interest^  and  no  part  of  the  interest  ever 
forms  a  part  of  the  principal  carrying  interest, 

COMPOUND  INTEREST  BY  DECIMALS. 
RULE 

MULTIPLY  the  given  prinvipal  continually  by  the 
amount  of  one  poosid,  uron*:  diJUi,  for  one  yeai^at  the 
rate  per  cent,  given,  until  the  number  of  multiplications 
are  equal  to  the  given  number  oC  years,  and  the  product 
will  be  the  amount  recjuired. 

Or,  In  Table  1.  Appendix,  find  tiie  amount  of  one  dol- 
lar, or  one  pound,  for  the  given  number  of  years,  which 
multiply  by  the  given  principal,  and  it  will  give  the 
tmount  as  before. 


17S  rN  VOLUTION. 


■VPLES. 


1.  Wh-;  to  in  4  ye^ii-u  at  G  per  cent. 

per:: 

^:504,99-f  or 

: /;oo-'i  ivs.  '^a.        ^   --  .^?25. 

';'.:c  :::Mne  bv  Table  1. 


V  amount - 

£.  Required  t!ie  aincunt  cf  425  dols.  .  3  years, 

at  6  per  cent,  coiripound  iriterest.        .1.  .7^cfs.-f 

5.  What  is  the  compound  interest  o^  5Ji'  dels,  for  14 
years,  at  5  per  cent.  ?     Bj  Table  L    Ans,  go-lo^secf^.-f 

4.  What  will  50  dollars  amount  to  ia  20  years,  at  6  per 
cent,  compound  interest  ?  t^ns,  gl60'35cts'.  &lm. 


^  INVOLUTION. 

Is  the  multiplying^  any  nimber  with  itself,  and  that  pro* 
duct  by  the  former  multiplier ;  and  so  on  5  and  ^^(i  several 
products  which  aris€  are  called  powers. 

Tlie  au)iiber  denoting  the  height  of  the  power,  Is  called 
the  index,  or  e:?ponent  of  that  pov/er. 

EXAMPLES 

What  is  the  5th  power  of  8  ? 
8  the  rooter  ist  po^.vcr. 
8 

64  =  2d  po^ver,  or  square, 
8 


519.  =  ZiX  power,  orxube. 


4096  =  4th  power,  or  biquadrate. 
8 


52768  =»  5th  power,  or  surselld.    Jim. 


-iwvOLUHON,    OR    EXTRACTION   OF    UOOTS.  179 

What  is  the  sqiiajc  ol  17,1  ?  .'his,  :02,4l 

What  is  the  square  of  ,085  ?  J./  .  ,«.:;:  25 

What  is  tlic  cube  of  25,4  ?     *  .;.-.-.    ' 
What  is  tlie  biquadrate  of  12  ? 

What  is  the  square  of  7^  ?  ^Li  ■          ,. 


EVOLUTION,  OR  EXTRAC'»(fN  OF  ROOTS. 

W  HEN  the  root  of  any  powei  is  required,  the  busi- 
ness of  finding  it  is  called  the  Extraction  of  the  Root. 

The  root  is  that  number,  whicli  bv  a  continual  multipli- 
cation into  itself,  produces  the  given  power. 

Although  there  is  no  number  but  what  will  produce  a 
perfect  power  by  involution,  jet  there  are  manj  numbers 
of  "xhiQii  precise  roots  can  never  be  determined.  But,  by 
the  li^eip  of  decimals,  we  can  approsimrite  towards  the 
root  to  any  assigned  degree  of  exactiiej^s. 

The  roots  which  approximate,  are  called  surd  roots, 
and  those  wliich  are  perfectly  accutate  are  called  rational 
roots. 

A  Table  of  the  Squares  and  Cubes  of  the  rdri£  digits. 


Roots. 

|1 

2 

s 

^ 

5 

6! 

T 

8| 

9 

Sq'iiares. 

11 

4| 

9 

IG 

25, 

S6| 

49 

64  1 

81 

Cubes. 

M 

i«! 

il7 

64 

i  i^irf 

-3  J  6 

343 

512  1 

729 

EXTRACTION  OF  THE  SQl^VKS   ROOT. 

Any  number  multiplied  into  itself  produces  a  square. 

To  extract  the  square  root,  i-  nlv  to  lind  a  {lumber, 
which  being  multiplied  into  itself.  duce  the  giveo 

number.  '  ^ 

RULE. 

1,  Distinguish  the  given  number  into  periods  of  two 
figures  each,  by  putting  a  point  over  the  place  of  units, 
another  over  the  place  of  hundreds,  and  Sv)  on  ;  and  if 
tliere  are  decimals,  poiut  them  in  tlie  same  manner,  from 
units  towards  the  right  band  ;  whicl«  noJn^^  siiow  tke 
Uttmber  of  figures  ti^e  r^ot  will   ro^lsif   »!'. 

S.  Fiud  tii«  gi-eatest  «(n:are  uunibcT  lu  the  tirst^  or  teft 

/ 


J  80         EVOLUTION,  OR  fiXTKACTiOM   OF   ROOTS. 

hand  period,  place  tlie  root  ot  it  at  the  riglit  hand  of  the 
given  number,  (after  the  iganner  of  a  quotient  in  division) 
for  the  first  figure  of  the  root,  and  the  square  number 
under  the  period,  and  subtract  it  therefrom,  and  to  the 
remaindei  briiig  down  the  iiexi  period  for  a  ^dividend. 

S.  Place  the  double  of  the  root,  already  found,  on  the 
left  hand  of  the  divklaad  for  a  divisor. 

4.  Place  such  a  figure  at  the  riglit  hand  of  the  divisor, 
and  also  the  same  figure  in  the  root,  as  when  multiplied  . 
into  the  whole  (increased  divisor)  the  product  shall  be 
equal  to,  or  the  next  less  tiian  the  dividend,  and  it  will  be 
the  second  figure  in  the  root. 

5.  Subtract  the  product  from  the  dividend,  and  to  the 
remainder  joio  the  next  period  for  a  new  dividend. 

6.  Double  the  figures  already  found  in  the  root,  for  a 
new  divisor,  and  from  these  find  the  next  figure  in  the 
root  as  last  directed,  avid  continue  the  operation  ift  the- 
same  manner,  till  you  nave  brought  down  all  the  periods. 

Or,  to  facilitate'  the  foregoing  Rule,  when  you  have 
brought  down  a  period,  and  formed  a  dividend,  in  order 
to  find  a  new  figure  in  the  root,  you  may  divide  said  divi- 
dend, (omitting  the  right  hand  figure  thereof,)  by  double 
the  root  already  found,  and  the  quotient  will  commonlv 
be  the  figures  sought,  or  being  made  less  one,  or  two,v^iri 
generally  give  the  next  figure  in  the  quotient. 

EXAMPLES. 

1.  Hequircd  the  square  root  of  141225,64. 
141225,64(375,8  the  root  Exactly  without  a  remaindcF  ; 
9  but  when  the  periods  belonging  to  any 

—  given  number  are  exhausteu,  and  still 

67,512  leave  a  remainder,  the  operation  may 

469  be  continued  at  pleasure,  by  alnn^xiflg 

.  periods  of  cyphers,  &c.  ] 

745)4325 
5725 


7508)60064 
G0064 


'}  tCtoamvc 


EVOLUTION,  •R  EXtllA«TION  OF  ROOTS^        kl  i 


2.  What  is  the 

square 

root  of  1296  ? 

*Mswers^ 
36 

5.  Of        

56644  r 

4.  Of        

0499025  » 

5.  Of        

36372961  P 

-;"    6(1  >^ 

6.  Of       

!84,4P 

<   •        K^,5?  + 

r.  Of 

9712,693809? 

9^,5  ^S 

8.  Of        

0,45569  ? 

56/3+ 

9.  Of        

,002916  ? 

,054     • 

10.  Of        

45? 

6,708-f 

TO  EXTRACT  THE  SQUARE  ROOT  OF 

VUL«IAR  FRACTIONS. 

RULE. 

Btduce  ihe  fraction  to  its  lowest  terms  ibr  this  and  all 
other  roots ;  then 

1.  Extract  the  root  of  th«  nuinerator  for  a  new  nume- 
rator, and  the  root  of  the  denominatt4>  for  a  new  denomi- 
nator. 

2.  If -the  frtctioa  be  a  surd,  reduce  it  to  a  decimal,  and 
extract  its  root. 

EXAMPLES. 

1.  What  is  the  square  root  of  -^f^^-  ?  ^ns.  -| 

2.  What  i#  the  square  yoot  of  -^%\^  ?  Jlns.  ^| 
5.  What  is  the  square  root  of  |||  ?  .^ns.  ^ 

4.  What  A  the  square  root  of  20^  ?  ^^s.  4i 

5.  WhatT  the  square  root  of  243^^  ?      ^^ns.  15 i 

SURDS. 

6.  What  is  the  squart  root  of  |^-?  Mb.  9I3«-f 

7.  What  is  the  square  root  of  4^|  ?  Ans,  ,7745 -f 

8.  Required  the  square  poot  of  56:t  ?  ^ns.  6,0207+ 


APPLICATION  AND  USE- OF  THE  SQUARE 
ROOT. 

P'roulem  I,  .A  certain  Gonefa]  r^s  ansinnjor  SlU'i 
men  ;  how  many  must  he  place  m  rank  am  Uq*  to  faun 
V  h  (» !n  into  a  giiua  re  ?       ., 


tTOo 


A         KVOLUTIOK,  OB.   EXlWoTION  OJf   ROOTS. 

.^   ^  RULE. 

^  Extract  the  square  root  of  the  given  number. 

^/5184=r2  Ans. 

i.    :l    A  certaifi  square  pavement  contains  20rSe 
' ; '   •  c:.^  ^ul't  CI  tJie  sariiC  si/.e  :  I  demaiid  how  many 

.    .v-:   .u  M.i,.  .:X  A::  side:  ?  v/2073G==:i44  dns^ 

K)H,  iix,   io  liiida  mean  proportional  between  two 
lumbers. 

RULE. 
Multiply  the  given  numbers  together,  and  extract  the 
square  root  of  the  product. 

EXAMPLES. 

What  is  the  mean pjoportional  between  18  and  72 ? 
72x18^1296,  and  v/1296=36  dns. 

pROB.  IV.  To  form  any  body  of  soldiers  so  tliat  thcj 
may  be  double,  triDle,  vkc.*^  as  many  in  rank  as  in  file- 
RULE. 
Extract  tiie  square  root  of  1-2,  1-3,  &c.  of  the  given 
.  imniutjf  of  iueiiu  and  that  will  be  the  number  pf  men  in 
^I"        r.  i!  double,  triple,  5cc.  and  theproduc?t  will  be  the 
.■Ilk.  * 

EXAMPLES. 

I  be  so  farmed,  as  tiiat  the  mimber  in 
:    ■;:'e  the  number  in  file. 
.-:.r^u.    i.  and  v^656l=81  in  fUe,  and  81x2 
U:  rank, 
-'"OB-r^V.    Admit  10  hhds.  of  water  are^ischarged 
;:^ii.  a  leaden  pipe  of  2^  inches  in  diamelW,  in  a  cer- 
•  iij.'ie;  1  demand  what  the  diameter  of  another  pipe 
>:»-  be,  to  discharge  four  times  as  much  water  in  the 
^  Hme. 

RULE, 
ijuare  the  given  drameter,  and  multiply  said  square 
i>y  the  given  proportion,  and  the  square  root  of  the  pro- 
duct is  the  answer. 

S^««2,5,  and  2^x2,5=6,25  square. 

4  given  proportion. 

y^'y.OOrrsS  iJKh.  dlam.  Ans. 


EVOLUTION,  OR  EXTli ACTION  ,0F   aoOTS.  183 

PiioB.  VI.  The  sum  of  any  two  numbers,  and  ihev 
products  being  given,  to  find  each  number. 
RULE. 

From  the  square  of  their  sum,  subtract  4  lir  • 
product,  and  extract  the  square  root  of  t^e  i.^ 
which  will  be  the  difference  of  tlie  tuo  r,-  „         j 

half  the  said  difterence  added  to  half  tie  ^n/^fV  p^^  tSf 
greater  cf  tl^e  two  numbers,  and  the  said  \w'f  mflli^*  ^ 
subtracted  from  the  half  sum,  gives  the  lessei  ^ix^iU<i^i. 

EXAMPLES. 

The  sum  of  two  numbers  is  43,  and  their  product  r 

442  ;  what  are  those  two  numbers  ? 

The  sum  of  the  numb.  43x43=1849  square  of  do 
The  product  of  do.     442x  4  =  1768  4  times  the  ^r- 

Then  to  the  ^  sum  of  21 ,5  rnruii/. 

-fand—  4,5  v'81=9  diff.  ofthf 

Greatest  numb.  26,0  ^  4^  the  h  diff, 

^Answers. 

Least  numb.  17,0  J 

EXTACTION  OF  THE  CUBE  HOOT. 

A  Cube  is  anj  number  multiplied  by  its  square. 

To  extract  the  cube  root,  is  to  find  a  number,  which, 
being  multiplied  into  its  square,  shall  produce  the  given 
number. 

RULE. 

\.  Separate  the  pven  number  into  periods  of  iluee  fig- 
ures each,  by  putting  a  point  over  the  unit  figure,  and 
every  third  figure  from  the  place  of  units  to  the  left.;  and 
if  tliere  be  decimals^  to  the  right. 

2.  Find  the  greatest  cube  in  the  left  liand  period,  and 
place  its  root  in  the  quotient. 

S.  Subtract  the  cube  thus  found,  from  the  said  period, 
and  to  the  remainder  bring  down  tlie  next  period,  calling 
this  tlie  dividend. 

4,  Multiply  the  suqare  of  the  qiiotient  by  SOO,  cM\jit 
it  the  divisor. 


184         EVOLUTION,   OR    KXTIIACTION    OF    KOOra. 

5.  Seek  how  often  tlie  tUvisoi^mav  be  had  in  the  divi- 
dend, and  place  the  vamlt  in  the  quotient :  then  multiplj 
the  divisor  by  this  hist  quotieiit  tigure,  placing  tlie  pro- 
duct under  the  dividend. 

6.  Multiply  the  foniier  quotient  figure,  or  %ures  bj 
the  square  of  the  last  quoiiejit  figure,  and  that  product  bv  . 
SO,  and  place  the  product  unikr  tho  last ;  then  under  these 
t^vo  products  place  the  cube  5;f  !■  o  !a,s(  r« i^oli en t  figure, and 

^fi*iH|i]-f together,  cal»L  ■  trahend. 

^  '7:mMrm  the  subtra  .  .    ^  idend,  and  to  ' 

the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend :  with  which  proceed  in  tii  same  manner,  till  the 
whole  be  finished.  • 

Note. — If  the  subtrahend  (found  bj  the  forego?  ng  rule) 
happens  to  be  greater  than  the  dividend,  and  consequent- 
ly cannot  be  subtracted  f  '  .  vou  must  make  the 
last  quotient  figure  oi^e  (  which  find  a  uew  sub- 

trahend, fby  the  rule  forc-;j, ;.  i^^;)  dud  so  on  until  you  can 
subtract  me  subtrahend   froia  tlka  dividend. 

EXAMPLES. 

1.  Required  the  cube  root\f  18399,744. 

18399,744(^26.4  Root,  dns, 
8 


2X2=4X500  =  1200)10399  first  dividend. 

7200 

6x6=S6x2=r£x50=rr2i60 

6x6x6=  216 

9576  1st  subtrahend. 
26x26==676x300=«202800)82Sr44  2d  dividend. 

811200 

4x4=l6x26=:416x50==:  12480 

4X4X4===    64 

825744  Sd  subtrahend. 


JBiyOtUTIOXj  on  EXTRACTION  OF  ROOTS.  18;:; 

Note. — The  foregoing  example  gives  a  perfect  root; 
and  if,  when  all  the  periods  are  exhausted,  there  happens 
to  be  a  remainder,  you  may  annex  periods  of  cyphers,  and 
continue  the  operation  as  far  as  you  think  it  necessary. 

2.  What  is  the  cube  root  of  205379  ?  59 

614125?  &"> 

41421736  .>  546      - 

146363,183.^  5%r 

'  2S,5036£9.J^  3,09 -f 

'  80,763  ?  4,3£  V 

,162771336.?  ,!A6 

JjQuQS^  i 34  r-  *"■ '"'  ^  - 
122615527232"^ 

1.  Find  by  trial,  a  oil  he  near  to  .'• 
call  it  the  supposed  cube. 

2.  Then  as  twice  the  supposed  cube,  addci}  tothegiv 
en  number,  is  to  twice  the  given  number  added  to  the 
supposed  cube,  so  is  th»  ioot  of  the  supposed  cube,  to 
the  true  root,  or  an  approximation  to  it. 

3.  By  taking  tlie  cube  of  the  root  thus  found,  for  the 
supposed  cube,  and  repeating  the  operation,  the  root  wili 
be  liad  to  a  greater  degree  of  exactness. 

EXAMPLES. 

Let  it  be  requii-ed  to  extract  the  cube  root  of  2. 

Assume  1,3  as' the  root  of  the  nearest  cube;  then— 
l,Sx.l,3xl,3==2,197=supposed  cube. 
'    'fhen,  2,197  2,000  given  number. 


4.394  4,000 

2,000  2,197 


As  6,394      :      G,197    :     :     K3     :      1,2599   root, 
which  is  true  to  the  last  place  of  decimals  ;  but  might  by^ 
repeating  the  operation,  be  brought  to  greatei  exactness,  1 
2.  What  is  the  cube  -oot  of  584^277056 


IS6  EVOLUTION,   OR   EXyilACTICK   OF  ROOTS. 

5.  Required  the  cube  root  cf  729001101  ? 

Ms,  ^0050094 

QUESTIONS, 

Showing  the  use  of  the  Cube  Root, 

1.  The  statute  bushel  contains  2150,425  cubic  or  solid 
fhclies.  I  demand  th5  fide  of  a  cubic  box,  which  shall 
contain  that  qufmtitj  ? 

f/21 50,425=12,907  inck.  dns. 
Note. — TliD  Solid  contents  of  si»rmlar  figures  are  in 
proportion  to  each  othm*,  as  the  cubes  of  their  similar 
sides  or  diameters. 

2.  If  a  bullet  3  inchef  diameter,  weigh  4lb.  what  will 
a  bullet  of  the  Sftme  metal  weigh,  w^hose  diameter  is  6 
inches?  ^^^ 

SxSx3=»t27  6x6x6=216  As  27  :  4lb.  t  :  216: 
52lb.  Ms. 

S.  If  a  solid  globe  of  silver,  of  3  inche?s  diameter,  be 
wortli  1 50  diOUars ;  what  is  the  value  of  anotlicr  globe  of 
silver,  whose  diameter  is  six  inched  r         S 

3x3x3=27  6x6x6=216  As  27  :  150  :  :  216  •: 
gl200.  Ms. 

The  side  of  a  cube  being  given,  to  faid  the  side  of  that 
cube  wich  shall  be  double,  triple,  &c.  in  (]fuantitj  to  the 
given  cube. 

RULE. 

Cube  your  ^/vcn  side,  and  multiply  by  the  given  pro- 
portion betwi?;en  the  given  and  required  cube,  and  tlic 
ciube  rof>t  of  the  product  will  be  the  side  sought. 

4.  If  a  cube  of  silver,  whose'side  is  two  inches,  be  worth 
20  dollars;  I  demand  the  side  of  a  cube  of  like  silvei;;, 
whtSe  value  shall  be  8  limes  as  much  ? 

2x2x2=8  and  8x8=64^64=4  inches.  Ms, 

5.  There  is  a  cubi«cal  vessel,  whose  side  is  4  feet:  I 
demand  the  side  of  another  cubical  vessel,  w'nich  shall 
contain  4  times  as  much  ? 

4x4x4=64  and  64x4=256^256«=6,549-j;A.  .^'^?. 
€i  A  cooper  Imving  a  cask  40  incl^?  lon'r,  ;:'''♦  f;;'  •''- 


,  ^  riOX    OF    ROOTS.  >  ";, 

c:h;^  ai  u>»^  liung  diainelei',  is  ojdered  ft)  make  anotlier 
cask  of  the  same  shape,  but  to  hold  just  twice  as  much  ; 
\vhat  will  be  the  bung  diameter  and  length  of  the  new 
cask  ? 

40x40x40x2=128000  then  ^3/ 128000 =50,3+  length. 
32x52x32x2=65o3(i  and  -^>'G55S6==40.3+&u?z^  dtam. 


J  General  Rule  for  Exivactin^  the  Roots  of  all  Powers, 
RULE. 

1.  Prepare  iJat  given  iiiin!]>er  for  extraction,  by  point 
ing  off*  from  tlie  unit's  place,  as  the  required  root  directs 

2.  Find  the  first  figure  of  the  root  by  triiil,  and  subtract 
its  power  from  tlie  left  hand  period  of  t'le  given  number. 

5.  Totlie  remainder  bring  down  tlie  first  fignre  in  the 
next  period,  and  call  it  the  dividend.   .J** 

4.  Involve  the  root  to  the  next  inferior  power  to  t'hat 
which  is  given,  and  multiply  it  bj  the  number  denoting 
the  given  power,  for  a  di^sor. 

5.  Find  how  many  times  the  divisor  may  be  had  in 
W\^  dividend,  and  ^^  c^uotient  will  be  another  figure  of 
the  root. 

6.  Involve  the  whole  root  ioih^  given  power,  and  sub- 
tract it  (always)  from  as  many  i^cj'iods  of  the  given  num 
ber  as  you  have  found  figures  in  the  root.  ^^^ 

r.  Bring  dovrn  the  first  figure  of  the  nQ::{i  period  tt)  the 
remainder  for  a  new  dividend,  to  wliich  find  a  new  divi- 
sor, as  before,  and  in  like  manr^cr  proceed  till  the  whole 
be  finished. 

Note. — Wlicn  the  number  to  be  subtracted  is  greater 
than  those^periods  from  which  it  is  to  be  ^  >.^:'  v  ^'^^  l-^-^t 
quotient  figure  must  be  taken  less,  &c. 

EXAMPLES. 

1.  Required  the  cube  root  of  135790,744  by  the  above 
I^Qiicral  melihod. 


^Sb  #    ?iYOX*uTmN,  on  ExyRACTioN-  v^ir  aoots. 


155796744(51,4  the  root. 
125=t=lst  subtrahend 


75) lOr  dividend. 

132651  =2d  subtrahend, 
rSOS)  S1457=2d  dividend. 

155r96r44=3d  subtraltend. 

5x5x3=75  iirst  divisor. 
51x51x51=132651  second  subtranend. 
51x51x3=7803  second  divisor. 
514x514x514=135796744  third  subtrahend. 

S.  Required  the  sursolidj  or  fifth  root  of  6436345. 
6436348'J23  root 


2X2X2X2X5=80)323  dividend. 
23X23X23X23X23=6436343  subtrahend. 

Note..— The  roots  of  most  powers  may  be  found  by  the 
square  and  cube  roots  only  5  therefore,  when  any  even 
power  is  given,  the  easiest  method  will  be  (especially  in 
a  very  high  power)  to  extract  the  square  root  of  it,  which 
reduces  it  to  half  tiie  given  pov*  er,  then  the  square  root  of 
that  power  reduces  it  to  half  the  same  pov/er  ^^and  so  on^ 
till  you  come  to  a  square  or  a  cube. 

For  example:  suppose  a  12th  power  be  given;  the 
square  root  of  that  reduces  it  to  a  sixth  power :  and  the 
square  rapt  of  a  sixth  power  to  a  cube, 

EXAMPLES. 

3.  What  is  llie  biquadrate,  or  41h  root  of  19957173576  i 

Jns.  37G. 

4.  Extract  tlie  square,  cubed,  or  6feh  root  of  1223059( 
464.  '  *^)is.  48. 

5.  Extract  the  square,  biquadrate.  or  Stli  root  of  72131 
95789358SS6.  Jlns.  96 


ALLIGATION, 

Is  the  mciliocl  of  mixing  several  siin|^!e3  of  liifterent  qual- 
ities, 90  that  the  composition  may  be  of  a  mean  or  middle 
quality ;  It  insists  of  two  kinds,  viz.  Alligation  Medial, 
and  Alligation  Alternate. 

ALLIGATION   MEDIAL, 

Is  wlien  the  quantities  and  prices  of  several  things  are 
given,  to  find  the  mean  price  of  the  mixture  composed  of 
'those  materials. , 

RULE. 

As  the -vvliole  composition  :  is  to  the  whole  value  :  :  so 
}s  any  part  of  tliji  composition  :  to  its  mean  price. 

EXAMTLES. 

1.  A  farmer  mixed  lo  busliels  of  rye,  at  64  cents  a 
bushel,  18  bushels  oF  Indian  corn,  at  55  cts.  a  bushel,  and 
21  bushels  of  oats,  at  28  cts.  a  bushel  5  I  demand  '^vhat  a 
bushel  of  this  mixture. is  v/orth.^ 

hu.       cts.     ^cts.      .  hu,     g  cts.         III. 
15  at  64=9,60    As  54  :  25,38  :  :  1 
18        55=^-9,90  1 

21       Ii8=5,8S  cii^. 

—  , 54)25.58(,4r  Answer. 

54  25,3S 

2.  If  20  bushels  of  wheat  at  1  dol.  S5  cts.  per  bushel, 
be  mixeil  with  10  busliels  of  rye  at  90  cents  per  bushel, 
what  will  a  bushel  of  tjiis  mixi'jve  be  wcrlli  r 

Jns.  St,  9.0cis. 

3.  A  Tobacconist  mixed  36  lb.  of  Tobacco,  at  Is.  6d. 
r  lb.  12  lb.  at  2s.  a  pound,  v/ilh  12  lb,  at  Is.  lOd.  per 

:b.5  what  is  the  price  of  :.  '  this  nuxture  ? 

Alls.  is.  Sd. 

4.  A  Grocer  mixed  2  C.  of  t-u<^ar,  at  56s.  per  C.  and  1 
C.  at  433.  per  C,  and  2  C.  at  50s.  per  C.  together  5  I  de- 
mand tlic  price  of  3  cwt.  of  this  mixture  ?  Ans.  £7  ISs. 

5.  A  Witie  merchant  mixes  15  gallons  of  wine  at  4s. 
2d.  per  gallon,  with  24  gall  or*  s  at  6s.  £d,  and  20  gallons, 
at  6s.  Su. ;  what  is  a  i>;ai!ou  of  tliis  composition  worth  P 

Ans.  5s.  IQd.  2f|grs. 


r,. 


6.  A  grocer  hith  several  sorts  of  sugar,  v'iz.  one  sort? 
p.t  8  ilols.  percwi.  an  other  sort  at  9  dols.  per  cwt.  a  third 
sortxit  lOtiols.  percvvt.  and  a  Iburth  sort  at  12  dols.  per 
cwt.  and  he  v.ould  mix  an  equal  quantityl)f  each  togeth- 
er; I  demand  the  price  of  3\  cwt.  of  this  mixture  ? 

\^ns,  S34  IQcts.  5m. 

7.  A  Goldsmith  melted  together  5  lb.  of  silver  bullion,  m 
of  8  oz.  fine,  10  It^.  of  7  oz.  fine,  and  15  lb.  of  6  oz.  fine ;  • 
pray  what  is  the  quality,  or  fineness  of  this  composition  i* 

Jlns.  6oz.  ISpwt,  ^gi\jini>r 

8.  Suppose  5  lb.  of  gold  of  22  carats  fine,  2  lb.  of  2!  | 
c;iratB  fine,  and  I  lb.  of  alloy  be  melted  together ;  what  is  1 
the  quality*  or  ilneness  of  this  n^.ass  ?  \ 

Jins.  19  carats  f.na 


ALLIGATION  ALTERNATE, 

IS  the  method  of  finding  what  quantity  of  each  of  the 
ingredients,  whose  rates  are  given,  will  compose  a  mix- 
ture of  a  given  rate ;  so  that  it  is  the  reverse  of  alligation 
medial,  and  may  be  proved  by  it. 

CASE.  I. 

When  the  mean  rate  ot  the  whole  mixture,  and  ihe 
rates  of  ail  the  ingredients  are  given  without  any  }in\ited 

quantity. 

"^  RULE. 

^  1.  Place  tliC  several  i-ates,  or  prices  of  the  simples,  be- 
ing reduced  to  one  denomination,  in  a  column  under  each 
other,  and  the  meaa  price  in  the  like  name,  at  the  left 
hand. 

2.  Connect,  or  link,  the  pnce  of  each  sim.ple  or  ingre- 
dient, v/hich  is  less  than  that  of  the  mean  rate,  with  one 
f)r  any  number  of  those,  whicii  are  greater  than  the  mcaa 
rdt^^  and  each  greater  rate,  or  price  with  one,  or  any 
number  of  the  less. 

S.  Place  the  difference,  betv/een  the  mean  price  (or 
mixtui^c  rate)  and  that  of  each  of  the  simples,  opposite 
to  tno  rntcs  with  which  thev  are  connected. 


ALLIGATION     ALA  iill.S  A  IE.  l9i 

4.  Then,  if  only  one  difference  stands  against  an^  r;ite, 
it  will  be  the  quantity  belonging  to  that  rate,  but  it  there 
be  itieveral.  their  sum  will  be  the  quantity. 

J^  EXAMPLES. 

1.  A  merclflnt  has  spices,  some  at  9d.  per  lb.  some  at 
^s.  some  at  S  and  some  at  2s.  6d.  per  lb.  how  much  of 
^.  ch  sort  must  lie  mix,  tliat  he  may  sell  tlie  mixture  at  Is 
8d.  per  pound  ? 

d.        d.    lb.  d         lb, 

f  9 ,10 at  9^.  .    r"9^        4*^  .^• 

d.JliTi        4     12  [^Glvesthe  d,\  12-f-^  10  [  g 

20  ]  24 J      I  8     24  r-'hisirer,  or  20  ]  24 J    j  H  f  g 


^ 


24J    j  11  j^l 
50 — ^     8j  '^S 


(^30_^ll     30  J  {J 

2.  A  gi*acer  would  mix  the  ibilov/ing  quantities  of  su- 
gar 5  viz.  at  10  cents,  13  cents,  and  16  cts.  per  lb.  :  what 
quantity  of  each  sort  must  be  taken  to  make  a  mixture 
worth  12  cents  ner  pound  ? 
'^na.  Sib.  at  lOcis.  ^Ib.  at  \Scts.  and  9lb.  at  16  cts.  per  lb. 

5.  A  grocer  has  two  sorts  of  tea,  viz.  at  9s.  and  at  155. 
per  lb.  how  must  he  mix  them  so  as  to  afford  the  compo- 
sition for  12s.  per  lb.  .^ 

Ans.  He  must  mix  an  equal  quantity  of  each  sort. 

4.  A  goldsmith  would  mix  gold  of  17  carats  fine,  with 
some  of  19,  21,  and  24  carats  line,  so  that  the  compound 
may  be  22  carats  tine  j  what  quantity  of  eacli  niubt  h^ 
take. 

^ns.  2  of  each  of  the  first  three  sorts^  and  9  of  the  last. 

5.  It  is  required  to  mix  several  sorts  oi  rum,  viz.  at  5s. 
.  and  9s.  per  gallon,  with  water  at  0  per  gallon  to- 

!Z:cthcr,  so  that  the  mixture  may  be  worth  6s.  per  s:allon  ; 
[low  much  of  each  sort  must  the  mixture  consist  of  .^ 
*ins.  1  gal.  of  Rum  at  5s.  1  do.  at  7s.  6  do  at  9s.  and  3 

fals.  water.     Or^  3  gals,  mm  at  5s.  6  do.  at  7s.  1 
o.  at  9s.  and  1  gal.  water. 

6.  A  grocer  hath  several  sorts  of  suf^ar,  vi/..  one  sort 
at  12  cts.  per  ib.  another  at  11  cts.  a  tiiird  at  9  Cts.  and  a 
fourth  at  8  cts.  per  lb. ;  1  demand  how  much  of  each  sort 
;i»ust  he  mix  together,  that  tlie  whole  <Ji;-u;(ily  may  bi> 
ifitfrdrd  at  10  &wU  ;;cr  noinni  - 


W^  ALTEIINATIOK  IPARTIAIii. 

^.      cts,  lb.     cts.  lb.    cts. 

f £  at  12  ri  at  12  fs  at  12 

Ist   Anq  ^  1  at  11     ,    .      J  2  at  11  ^  ,    ^^  J  2  at  11 

Uat    8  Llat    8     IB       tsat  8 

4th  Ans.  Slh,  of  each  sort.'*  w 

CASE  11. 

ALTERNATION  PARTIAL. 

Or,  when  one  of  the  ingredients  is  limited  to  a  certain 
quantity,  tlience  to  find  the  several  quantities  of  therest> 
in  proportion  to  the  quantity  given. 

RULE. 

Take  the  difterenee  between  each  price,  and  the  mean 
rate,  and  place  them  alternately  as  in  Case  L  Then,  as 
the  difference  standing  j^ainst  that  simple  whose  quantity 
IS  given,  is  to  that  quantity  :  so  h  each  of  the  other  dif- 
ferences, severally,  to  the  several  quantities  required. 

EXAMPLES. 

1.  A  farmer  would  mix  10  bushels  of  wheat,  at  70  cts. 
per  bushel,  with  rye  at  48  cts.  corn  at  SQ  cts.  and  barley 
at  30  cts.  per  bushel,  so  that  a  bushel  of  the  composition 
ma  J  bci^old  for  58  cents ;  what  quantity  of  each  must 
be  taken. 

rzo- — ^  8  sptands  against  the  given  quan« 
Moan  .ate,  SsJg     |  f,  ftity 

[so i  3-2 


As  S  :  10 


F  2  :    2^^  bushels  of  rye. 
;  •<  10  :  12 i  bushels  of  corn. 
(^32  :  40    bushels  of  barley. 


*  Th^se four  answers  arise  from  as  nmny  various  ivaij^ 
of  linking  the  rates  of  the  ingredients  together. 

Questions  in  this  rule  adrdit  cf  an  infinite  variztif  of  an- 
swers :  for  after  the  quantities  are  found  from  different 
inetkods  of  Unking ;  any  other  numbers  hi  the  samevropor- 
tion  between  themselmSyaSithemmioers  which  compose  the 
^nswor^  imll  Jifce;t%^5  saii0 the  conditions  ofih^i^u^tion 


ALiKRNATlON    PARilAA-  AiSh 

2.  How  much  water  miist  be  mixed  with  lOO  gallons 
of  rum,  worth  7s.  6d.  per  gallon,  to  reduce  it  to  6s.  Sd. 
per  gallon  ?  Jim,  20  gallons, 

S,  A  farmer  would  mix  20  bushels  of  rye,  at  65  cents 

Eer  bushel,  witii  barley  at  51  cts.  and  oats  at  30  cts.  per 
ushel ;  how  much  barley  a?ul  oats  must  be  mixed  with 
the  £0  bushels  of  rye,  that  the  provender  may  be  worth 
41  cents  per  bushel  ? 

^ns.  20  bushels  of  bavhij^  and  61^  bushels  of  oats. 
4,  With  95  gallons  of  rum  at  8s.  per  gallon,  I  mixed 
other  rum  at  6s.  8(1.  per  gallon,  and  some  water;  then  1 
found  it  stood  me  in  6s.  4d,  per  gallon  5  I  demand  how 
much  rum  and  how  much  water  I  took  ? 

^^ns.  95 gals,  rum  at  6s,  S(L  and  SO  g^als,  water. 

CASE  III. 

When  the  whole  composition  is  linl^ted  to  a  given  quantity. 

RULE. 

Place  t:LC  diflerence  between  tlie  mean  rate,  and  the 
several  prices  alternately,  as  in  Case  I. ;  then,  As  the 
sum  of  the  quantities,  or  diflerence  thus  determined,  is  to 
the  given  quantity,  or  whole  composition  :  so  is  tiie  diife- 
renceof  each  r^Ucjto  the  required  (quantity  of  each  rate. 

EXAMI'LES. 

1,  A  grocer  had  ("our  sons  of  tea,  air  Is.  5s.  6s.  and  10s. 
per  lb.  tiie  worst  would  not  sell,  and  the  best  were  too 
dear;  he  tluuefore  mixed  UMilb.  and  so  much  of  each 
sort,  as  to  sell  it  at  4s.  per  lb.  5  how  nuich  of  each  sort  did 
ne  take  ? 


>V^T 


s. 

lb. 

Vk 

1 — ^ 

6 

"6 

S^ 

2          Ih.     Ik 

6J 

I  As  12  :  12Q  ; 

:^  1 

10-— J 

3 

^3 

60 

at 

1-^ 

20 

—. 

S 

10 

«» 

6 

30 

.^ 

10 

Sum,     12  120 

i[7 


.■^.i  AIUTHMETICAL    rKOGKESSION. 

2.  Hcrvy  much  water  at  0  per  galloiu  must  be  mixed 
with  wine  at  90  cents  per  gallon,  so  as  to  iill  a  vessel  of  100 
gallons,  which  may  be  ailbrded  at  60  cents  per  gallon  ? 
^n's.  334  gals,  ivater^  and  6o-|  gals,  ivine. 
S.  A  grocer  having  sugars  at  8  cts.  16  cts.  and  24  cts. 
per  pound,  would  make  a  comptisition  of  240  lb.  worth 
£0  cts.  per  lb.  without  gain  or  loss  ;  what  quantity  of  each 
must  be  taken  ? 

^ns.  40  lb.  at  8  cts.  40  (d  16  cts.  and  I16O  at  24  cts, 

4.  A  goldsmith  had  two  sorts  of  sliver^  bullion,  one  of 
10  oz.  and  the  other  of  5  oz.  line,  and  has  a  mind  to  mi:. 
a  pound  of  it  so  that  it  shall  be  8  oz  fine  5  how  much  oi' 
each  sort  must  he  take  s' 

Jins.  4|  of  5  oz.fne,  and  7^  of  \0  oz.fine. 

5.  Brandy  at  Ss.  6d.  and  5s.  9d.  per  gallon,  is  to  be 
mixed,  so  that  a  lihd.  of  63  gallons  may  be  sold  for  If:;. 
12s.  5  how  many  gallons  must  be  taken  of  each  ? 

Jihs.  14  gals,  at  5s.  9  J.  and  49  gals,  at  Ss.  6d. 


ARITHMETICAL  PROGRESSION. 

Any  rank  of  numbers  more  than  two,  increasing  by 
common  excess,  or  decreasing  hv  common  difierence,  is 
said  to  be  in  Arithiii«itical  Progression. 

<;,    C  2,  4,  6,  8,  &c.  is  an  ascending  arithmetical  series : . 
I  8,  6,  4,  2,  &:c.  is  a  descending  arithmetical  series  : 

The  numbers  which  form  the  series,  are  called  the 
\  erms  of  the  progression ;  the  first  and  last  terms  of  which 
are  called  the  extremes.* 

PROBLEM  I. 

The  first  term,  the  last  term,  and  the  number  of  terms 
being  given,  to  find  the  sum  cf  ail  tlie  terms. 


*A  series  in  progression  includes  five  vutcs^  viz.  the 
first  term,  last  teriUj  number  of  terms^  common  difference^ 
and  sum  of  the  series. 

By  having  any  three  of  these  parts  given,  the  other  two 
may  be  fourui',  ivJiich  adraitsof  a  ^^  -.'/-'  *  of  rrohlems  ;  bu* 
most  of  them  are  best  inukvstou  'i!:€h'aic  "pr-aces':^, 

an^  arc  ■^^^^"  'y'^itied* 


HULK. 

Multiply  tlie  sum  of  tha  extremes  by  the  number  of 
terms,  and  half  the  [yroiluctwill  be  the  answer. 

KXAMTLES. 

1.  The  first  term  ol"  an  arithmetical  series  h  3,tiie  last 
term  ^3,  and  the  number  of  terms  1 1  :  recpjircd  tlic  sum 
of  the  series 

^3+3=26  Slim  of  the  extremes. 
Then  26x11-^^^143  the  Answer. 
^2,  IIov/  many  strokes  clue^  tlie  hammer  of  ;i  c^ock 
strike,  in  twelve  hours  ?  *f}ns.  78 

3.  A  merchant  sold  100  y;!i<ls  <  viz.  the  first 
yard  for  let.  the  second  for  2  cts.  me  nwva  for  5  cts.  Sec* 
1  demand  \vliat  t\\^  cloth  came  to  at  that  rate  ? 

Ms.  S50^. 

4.  A  man  bois^lit  19  yards  ofjinen  in  arithmetical  pro- 
gression, for  the  iirst  yard  }\egave  Is.  and  for  the  last  yd. 
IL  irs.  what  did  the  vv!ir*le  come  to?      *B.ns.  £18  1^. 

5.  A  draper  sold  100  yds.  of  broadcloth,  at  5  cts.  for 
the  first  yard,  10  cts.  for  the  second,  15  ii)v  the  third,  &c. 
increasing  5  cents  for  ^\tvj  y^viX  :  What  did  the  whole 
amount  to,  and  wliat  did  it  avera^^c  per  yard  P 

M^,  Ji^T.Gurit^  S252i,  and  the  average {irice  is  §2, 52cts. 
5  mills  per  yard. 

6.  Suppose  144  oranges  were  hdd2yards  distant  from 
each  otlier,  in  a  right  line,  and  a  basket  placed  two  ynnls 
from  the  first  i>range,  whatjength  <>f  «i;rotind  will  tliat  boy 
travcl  over,  v/ho  gathers  tltem  nn  Finclv,  leturnin^';  witli 
them  one  by  one  to  the  br^ske-  ' 

'Bus. 'IS  m.      ,  .  .  ^;:;,  ISO  lids. 

PROBLEM  11. 

The  first  term,  ti»e  last  term,  and  the  number  of  terms 
given,  to  find  the  common  tlifference. 

RULE. 

Divide  the  dill'erence  ol  ihe^  extremes  by  the  number 
of  terms  less  1,  and  tae  quotient  ^v?n  i^o  the  cianmon  dif- 
ference. 


19^  ARlTHMETrCAJ.    I'UQGHESSION. 


1 


EXAMPLES, 

1.  The  extremes  are  3  and  S9)  and  tlie  number  ol 
terms  14,  what  is  the  common  diiibrence? 

'"^  i  Extremes. 

Number  of  terms  less  1=13)26(2  ^ns. 

2.  A  man  had  9  sons,  whose  several  ages  diffeiftd  alike, 
the  youngest  was  3  years  old,  and  the  oldest  55;  what 
was  the  common  diilerence  of  their  ages  ? 

Jifis.  4  years. 

3.  A  man  is  to  travel  from  Kew-London  to  a  certain 
place  in  9  days,  and  to  go  but  3  miles  the  first  day,  in- 
creasing every  day  by  an  equal  excess,  so  that  the  last 
day's  journey  m?.j  be  43  miles  :  Required  the  daily  in- 
crease, and  the  length  of  the  whole  journey  ? 

^ins.  The  daily  increase  is  5,  and  the  whole  jmvney 
207  miles, 

4.  A  debt  is  to  be  discharged  at  16  difterent  payments 
(in  arithmetical  progression,) the  first  payment  is  to  be 
lAl.  the  last  lOOZ. :  What  is  the  common  difference,  and 
the  sum  of  the  whole  debt  ? 

Ans,  5L  I4s.  St/,  common  dlference^  and  9121,  thewhoh 
debt, 

PROBLEM  III. 

Uiven  the  first  term,  last  term,  and  common  difference,. to 
ill  1 1  the  number  of  terms. 

l^ULE. 

Divide  tlie^ difference  of  tlie  extremes  by  the  common 
diffei-ence,  and  the  quotient  increased  by  1  is  the  number 
of  Icrms. 


EXAMPL] 


1.  If  tlie  extremes  be  3  and  45,  and  the  common  dif- 
ference 2 ;  what  is  the  number  of  terms  ?         %dns>  22. 

2.  A  man  going  a  journey,  travelled  the  first  day  five 
miles,  the  last  day  45  miles,  and  each  day  increased 
his  journey  by  4  miles:  how  many  days  did  he  travel  j 
and  how  far  ? 

Jr.?.  11  days^  and  thf*:  'wh-l  '  travelled  ^75  miles. 


JE^I 


\ 


geSKietrioal  progressiov.  197 

GEOMETRICAL  PROGHESSION, 

IS  V,  hen  any  rank  or  series  of  numbers  incioased  by  one 
common  multiplier,  or  decreased  by  one  common  divisor  , 
as  1,  2,  4,  8,  16,  &c.  increase  l)y  the  multiplier  Sj  and 
£7,  9,  3,  1,  decrease  by  the  divisor  3. 

PROBLEM  I. 

The  first  term,  ih<i  last  term  (or  tlie  extremes)  and  the 
ratio  given,  to  find  the  sum  of  the  series* 

RULE. 

Multiply  tlie  last  term  by  the  ratio,  and  from  the  pro- 
duct subtract  tlie  fust  term  ;  then  divide  the  remainder 
by  the  ratio,  less  by  ],aud  tlie  quotient  will  be  tlie  sum 
of  all  the  terms. 


1.  If  the  series  be  2,  6,  18,  54,  \G9..  486.  I^.-).^.  nnd 
the  ratio  S,  what  is  its  siiui  total  ? 

SX145S-— 2 

" =2186  thc.^nsiver. 

2.  The  extremes  of  a  geometrical  seiics  are  1   iind 
65536,  and  the  ratio  4 ;  what  is  the  sum  of  tlie  sc-ics  .^ 

Jlns.  srnbi. 

PROBLEM  IL  / 

Given  the  first  term,  and  the  rati(?,  to  find  any /ther  term 
assigned.*  "/ 

CASE  L 

When  the  first  term  of  tlie  series  and  the  ratio  are  equahf 


*As  the  last  term  in  a  long  serks  of  numbers  is  very  fe- 
dioiis  to  he  found  by  continual  inidtipliccilions^  it  ivill  he 
necessary  for  the  readier  finding  it  ont^  to  have  a  serie:^ 
of  numbers  in  arifhmetical  vroportion^  called  indices^ 
U'hor.p  cnminnnciijjvrence  is  I. 

t  'r/  '.n  the  first  term  of  the  :ierie^nd  the  ratio  are  equals 
Ihe  i/i'lices must beginivith the  unity  and  in  this  case^  t}i$ 


n» 


4n-    c 

?93  GEOMV.TRIOAL    FROGIIESSION* 

1.  Write  do^?n  a  few  ot  tlic  leadiwg  terms  of  the  se- 
ries, and  place  their  indices  over  them,  beginning  the 
indices  with  an  unit  or  1. 

2.  Add  together  sudi  indices^  whose  sum  shall  make 
up  the  entire  index  to  tlic  sum  required, 

5.  Multiply  "Ai^  terms  of  the  geometrical  series  belong, 
ing  to  those  indices  together,  and  the  product  will  be  the 
it^xa.  sought. 

EXAMPLES, 

1.  If  the  ilrstbe  %  and  tlie  ratio  £;  Vviiat  is  the  13tk 
term. 

1,  2,  S,     4,     5,  indices.  Ih^*"   '"   -  '  -™1S 

2,  4,  8j  16j  S2,  leading  terms.     Z'-:  'I  ^ns. 

2.  A  draper  sold  20  yards  of  .superfine  cloii),  the  first 
yard  for  Sd.  the  second  for  9d.  the  tliird  for  ^7d,  &c.  in 
triple  proportion  geometrical  x  what  did  the  cloth  come 
to  at  that  rate  ?         ■  •  ' 

The  SOth,  or  L'lst  term  is  3486784401^. 
Then  3+3486784401—3 

■ ==5230176600f/.  the  sum  of  all 

the.terms  (by  Frob.  I.)  equal  to  £21792402  10s.  Jns. 

S.  A  ricli  miser  thought  20  guineas  a  price  too  much 
for  12  fine  horses^  but  agreed  to  give  4  cents  for  the  first, 
16  cents  for  the  second,  and  64  cents  for  the  third  horse, 
and  so  on  in  quadruple  or  four-fold  proportion  to  thelasl^ 
w^hat  did  "they  come  to  at  that  rate,  and  how  much  did 
tliey  c©st  per  head,  one  with  another  ? 

Sns,  Tlie  12  horses  came  to  g223696,  20ci,9.  and  the 
average  price  uas  gl8641,  ^5ct$,  per  head, 

product  of  avj^  fivo  terms  is  equal  to  that  teriiu  signi^ 
by  the  sum  of\their  indites, 

rpj         C123    4    5  S(c,I%dizes  or  arithmetical  seriei 
-itius,   ;^£  4  s  16  32  ^V.  ^eomeirical' series. 
JS'*(nvy    3+2  r^.  5  rrz^theindea'of  tkffftktermj  and 
4x8  »«  32  «  ike  ffthterm  , 


GEOiMETRlCAL    PHOGHESSION  .       199. 

CASE  II. 
When  the  first  term  of  the  series  and  the  'ratio  are  diffe- 
rent, that  is,  when  the  first  term  is  either  greater  or 
less  than  the  ratio.* 

1.  Write  down  a  few  of  the  leading  terms  of  the  series, 
and  begin  the  indices  with  a  cjpiier;  Thus,  0,  1,2,  S,  &c. 

2.  Add  together  the  most  convenient  indices  to  make 
an  index  less  by  1  than  the  niraiber  expressing  the  place 
ofthtf  term  sought. 

S.  Multiply  tlie  terms  of  the  geometncal  series  to- 
getli«r  belonging  to  those  indices,  and  make  the  product 
a  dividend. 

4.  Raise  the  first  term  to  a  power  wliose  index  is  one 
less  than  the  number  oT  the  terms  multiplied,  and  make 
the  result  a  divisor. 

5.  Divide,  and  the  quotient  is  the  term  sougl^t. 

F.XAMPI,I<:S. 

4.  If  the  first  of  a  geometrical  scries  he  4,  and  the  ratio  . 
S,  what  is  th.«a  7th  term  ? 
0,     1,     2,      3,  Indice.i. 
4,  12,  S6,  108,  leading  terms. 

5-f  2+1=6,  e(\^.  index  of  liie  7i\\  lers.*. 
108X36X12=46656,, 

r=291'6  the  7th  term  required. 

16 
Here  the  number  of  terms  multiplied  are  three;  there- 
fore the  first  term  raised  to  «t.  povv-er  less  than  three,  is  the 
2d  power  or  square  of  4  =  16  the  divisor. 

*  When  the  first  term  of  the  series  and  the  ratio  are  dif' 
ferent J  the  indices  mustbe^in  with  a  ojpher,  and  tM  sum 
of  the  indices  made  choice  of  virviilhe  one  less  tJian  the  num- 
ber of  terms  given  in  the  'question:  because  1  in  the  indices 
stands  over  the  second  te^rm^  and  2  in  the  indices  over  the 
third  term,  ^"c.  and  in  this  case,  the  prodnct  of  any  tivo 
terms,  divided  bij  the  firsts  is  equal  to  thai  term  beyond  the  \ 
Jirstj  si;:^nifie'f  hj-  the  siim  of  their  indices^ 
.„.^^.    <;o,.  r,  2,    S,    4,' iyc.  Indices. 

'^"  ■}  T,  3,  9,2:',  PI,  ^'c.  Geometrical  series. 
;      >  .   :- ^3 ^7  the  index  of  the  Wi  term, 

.    <  irvr^c'iar  the  Sik  thmiy  or  tJie  7th  beyond  tJiA-lsL 


200 


rosiTioN. 


5.  A  GoklsiniCii  sold  1  lb.  of  gold,  at  2  cents  for  tlie 
first  ounce,  8  cents  for  the  second,  32  cents  for  the  third, 
&CC.  in  a  quadruple  proportion  geometrically ;  what  did 
the  whole  come  t(j  ?        '  /Ins.  ^11184^,  lOc/s. 

3.  Yv'iiat  debt  can  be  disc]5a!>';ed  ni  a  year,  by  paying 

1  farthing  the  first  nr.  *    '     ' ''  "  rthings,  (or  2^d.)  the  se- 
en nd,  and  so  ou.  eiich  '-        old  proportion  ? 
."ills.  '      ^40  U^..9d.  Sqrs. 

r.  A  tlircshcr  worked  :':.:: a  fanner,  and  receiv- 
ed for  tlie  tlrst  day-s  \v{)rk  four  i>ariey-corns,  i\tr  the  second 
12  barley-corns,  for  tne  tiiird  36  barley-corns,  and  so  ob 
in  triple  proportion  geometrical.  I  demand  \vhat  the  20 
days'  labor  came  to,  supposing  a^pint  of  barley  to  contain 
7680  corns,  andtlKi  whole  quantity  to  i}e  sckl  at  2s.  6d, 
per  bushel  ?    ^ns.  £177 o  7s,  6d.  rejecting  remainders. 

8.  A  man  bought  a  horse,  and  by  a,Q;reeu)ent  was  i& 
give  a  farthing  for  the  first  nail,  two  fur  iha  second*  four 
tor  the  tiiird,  ckc.  There  were  four  shoes,  and  eight  nails 
in  each  shoe ;  what  did  the  horse  come  to  at  that  rate  ? 

Ans.  /j 4473924  5s.  S^d. 

9.  Suppose  a  certain  body,  put  in  motion,  should  move 
the  length  of  one  barley-corn  the  first  second  of  tlmv^^  one 
inch  the  secon<i,  and  three  inches  the  tliird  second  of 
time,  and  so  continue  to  increase  its  motion  in  triple  pro- 
portion geometrical ;  how  many  yards  would  the  said 
body  move  in  the  term  of  half  a  minute  ? 

.^ns.   953199685623  j/t/s.  Iff.  lin.  Ih.c.  which  is  no 
kss  \hanfive  hundred  and  fart  i/ -one  rrdllionsof  miles. 

POSITION. 

1  OSITION  is  a  rule  wliich,  by  false  or  supposed  num- 
bers, taken  at  pleasure,  discovers  the  true  ones  required. 
It  is  divided  into  two  parts,  Single  or  Double. 

SINGLE  POSITION, 

Is  when  one   number  is  required^  the  prOjpertios  of 
are  given  i^itb©  question. 


<i  IN  O  I.K    POSIT  low.  CU  » 

[iULE. 

1.  Take  any  number  and  perform  tiie  same  operation 
with  it,  as  is  described  to  be  performed  in  the  question. 

2.  Tfeen  say ;  as  tiie  result  of  the  operation  :  is  to  the 
given  sum  in  the  question  :  :  so  is  the  supposed  number  : 
to  tlie  true  one  required. 

T^he  method  of  proof  is  bj  substituting  the  answer  in 
the  question.  Kj 

EXAMPLES.  ^^ 

■J.  A  schoolmaster  being  asked  how  many  scholars  h^^ 
had,  said,  If  I  had  as  many  more  as  I  now  have,  half  a|( 
many,  one-tliird  and  oncrloiJ  rth  as  many,  I  sliould  theifti 
have  148 :  How  many  scholars  had  he  ?  ^  • 

Suppose  he  luid  ^12     As  S7  :  148  :  :  12  :  48  ^«i* 

24    r 

16         •^ 

Eesiilt,    S7  Proof,  148         ^ 

2.  What  number  is  that  v;i]ic]\  being  increased  by  4,  J,^ 
and  i  of  itself,  the  sum  will  be  125  ?  ^^ns,  60, ' 

S.  Divide  93  dollars  bct^v^en  A,  B  and  C,  so  that  B^^s 
share  may  be  half  as  much  as  A^s,  and  C's  share  three  i 
ti  jiies  as  much  as  B's. 

Jliis,  .r:>  share  51,  ^'s  152,  and  (Ps  461  dolls, 

4.  A,  B  and  C,  joined  their  stock  and  gained  360  dols, 
of  v.hich  A  took  up  a  certain  sum,  B  took  3|  times  as 
much  as  A,  ant!  C  took  up  as  much  as  A  and  B  both  5 
M'liat  share  of  ihi'^-  gain  had  each  ? 

Jr.>^,  .i  S40,  B  S140,  and  C  S18D. 

5.  Delivered  to  a  banker  a  certain  sum  of  money,  to 
receive  interest  for  the  same  at  G/,  pei^  cent,  per  annum, 
simple  interest,  and  at  tlie  end  of  twelve  years  received 
rsiZ.  principal  and  interest  together  :  What  was  the  sum 
delivered  to  him  at  first  ?  *8ns.  £425. 

6.  A  vessel  has  3  cocks,  A,  B  and  C  ;  A  can  till  it  m 
1  hour,  B  in  9  hours,  and  C  in  4  hours  5  in  what  time^vili 
they  all  fill  it  to»ether  ?  JIvs.  SAmin,  ITlsec, 


as  many 

=2 

12 

i 

as  many 

= 

G 

^ 

as  many 

= 

4 

i  as  many 

= 

0 

'1 ,. 

sitions  oT  i,. 

liUi/:. 

^             -  two  suppi)-  ' 

1^  Take  any  1 

wo  (■ 

.:,,-(->*      • 

3,  and  proceed ; 

v/it\i  eacii  accord 
,     2.  Find  iio^v  111 

uch  - 

:  ::  13. question. 
ll.ent  IVom  the 

^  A'osults  hi  the  queslluii. 

fW^    3.  Multiply  the  first  posltioi]  ;  . ^.-t  error,  and  ilie 

f  -Ixist  position  by  the  first  error. 
^  4.  If  tliG  errors  are  alike,  divide  the  diirerence  of  the 
^)r(iuucis  by  the  diuercnc  of  the  errors,  and  the  quGticnt 

J  ^willbe  theliiisvrer. 

^     0.  If  the  erro]s  ai-e  iinlikcj  divide  the  sum  of  the  pro- 
'ducts  by  tlie  sum  of  tlie  errors^  and  ""dxt  quotient  "will  be 
<the  answer. 
..,•    Note.  The  errors  arc  said  to  b?  ■'''-"  -.hen  they  are 

Y   both  too  gi-eat,  oi-botfi  too  small :  ?,  vvlien  one  ' 

^  ^  too  great,  and  "^'^  ^  -'' '^:"      ^   . 

• 

^[    1.  A  purse  of  100  dollars  is  to  be  divideji  among  4 

r     men,  A,  l^^C  and  1).  ^;>  th.r  ^*  iimv  I'^ive  4  dollars  more 
•    than  A,  and  C  8  d  .id  D  twice  as 


the  money? 

Suppose  A     8 
B  12 
C  20 
I)  40 

80 
100 

70 

1  Of  I 


ist.  error    50  9A.  error    20 

*Tkose  qucaiions^  in  ivhich  the  results  are  not  propor- 
tional to  their  positions^  belong  to  this  rule  ;  such  as  those^ 
in  ivhich  the  nitmher  sought  is  increased  or  diminished  hij 
som» given  number^  ivhich  is  no  known  part  rfthemmiber 
required 


BOUBLE    POSITION.  203  >^ 

The  errors  being  alike,  arc  both  too  small,  ther'^fore, 


Pos,         Err, 
6  SO 

2Il 


S 

'A 
B 

12 

16 

24 

^D 

48 

Proof, 

100  . 

8  20 

£40  120 

120 

10)120(12  A'spait 

2.  A,  Baud  C, built  a  house  which  cost  500  dollars, 
of  which  A  paid  a  certain  sum  ;  B  paid  10  dollars  more 
ihan  A,  and  C  paid  as  much  as  A  and  B  both  5  how  much 
did  each  man  pay  ? 

Jins,  Jiimul  120,  S  lSO,awfZ  C^50dols. 

5.  A  man  bequeathed  lOOL  to  three  of  his  friends,  aftei 
this  manner :  the  first  must  have  a  certain  portion ;  the 
second  must  have  twice  as  mucli  as  the  first,  wanting  8^ 
and  the  tliird  must  have  three  times  as  much  as  the  first, 
wantins;  15l.i  1  demand  how  much  each  man  must  liavc  ? 

dns.^The  first  £Q0  10s.  second £S5,  third  £46  10s. 

4.  A  laborer  was  hired  for  60  days  upon  this  condition ; 
that  for  every  day  k«  v.  roiiglit  he  sliould  receive  4s.  and 
for  every  day  he  was  idle,  should  forfeit  23. :  at  the  ex- 
piration of  the  lime  he  received  7L  10s. ;  how  many  days 
did  he  work,  and  liow  many  v/as  he  idle  ? 

^ns.  He  wrought  45  days^  and  ivas  idle  \  5  days. 

5.  What  number  is  tliat  which  being  increased  by  its 
i,  its  i,  and  18  more,  will  be  doubled  }  Jins,  72. 

6.  A  man  gave  to  his  three  sons  all  his  estate  in  money, 
viz.  to  F  half,  wanting  50Z.  to  G  one-third,  and  to  H  tile 
rest,  which  was  XOt'Jess  than  the  share  of  G5  I  demand 
iho^  suuj  given,  and  each  mari'spart.^ 

Ans,  The  sum  giv.en  was  £S60.  ivhereof  F had  £lSOf 


i04  PERMUTATION    OF    QUANTITIES. 

7.  Two  men  J  A  and  B,  lay  out  equal  sums  of  money 
Jji  trade ;  A  gains  126/.  and  B  looses  S7L  an'd  A's  money 
is  now  double  to  B's  :  what  did  each  lay  out  ? 

Jns.  £500. 

8.  A  farmer  having  driven  his  cattle  to  market,  reciv- 
ed  for  them  all  ISOZ.  being  paid  for  every  ox  7/.  for  every 
cow  SL  and  for  every  calf  1/.  10s.  there  were  twice  as 
many  c^wsasosen,  and  thn«e  times  as  many  calves  as 
cows  5  how  many  were  there  of  eacli  sort  ? 

Ans,  5  oxen^  10  coivs^and  SO  calves, 

9.  A,  B  and  C,  playing  at  cards,  staked  S24  crowns ; 
but  disputing  about  tricks,  each  man  took  as  many  as  he 
could :  A  got  a  certain  number;  E  as  many  as  A  and  15 
more  5  C  got  a  fifth  part  of  both  their  sums  added  togeth- 
er:  how  many  did  each  get  ? 

Jns.  .d  isri,  B  142^,  C  54. 


PERMUTATION  OF  QUANTITIES, 

Is  the  showing  hov/  many  diiTerent  ^vays  any  given 
number  of  things  may  be  changed. 

To  find  the  number  of  Permutations  or  chanc:es,  that 
can  be  made  of  any  given  number  of  tilings,  all  different 
from  each  other. 

liULi:. 

Multiply  all  the  terms  of  the  natural  series  of  numbers, 
from  one  up  to  tlie  ^ven  number,  confinually  together, 
and  the  last  product  will  be  the  answer  retjuired. 

EXAMPLES. 


1.  How  many  changes  can  be  f  1 

made  oi  tlic  three  first  letters  of  j  ^3 

the  alphabet?  Proof,  <j 


a  b  c 
a  c  b 
b  a  0 

4  b  c  a 

5  c  h  a 
1x2x3=^6  Jvifi,                      {6    cab 

2.  Ilovf  ifi'iXny  Atinr^^B  iViay  be  iMng  &n  9  bells  ? 


AXKiriTIES   OR   PENSIONS.  £05 

S.  Scvci\  gentlemen  met  at  an  mn,  and  were  so  well 
pleased  witli  their  host,  and  with  each  other,  that  thej 
agreed  to  tarry  so  Inn*^  as  t!ioy,  together  with  their  host, 
could  sit  every  day  in  a  dilVerent  position  at  dinner  ;  how 
long  must  iUi^y  have  staitl  at  said  inn  to  liave  fuliilled 
then- agreement  ?  JIns,  liO^l-^^  years. 


ANNUITIKS  Ofl  PKiXSlONS, 

COMrUTED    AT 

CASE  r.  ^'  -^^-^^^ 

To  find  tlie  amount  of  an  annuity,  or  Pension,  in  an'cars, 
at  Compound  Interesi:. 


RULE. 


^l^<^  ^fj 


1.  Make  1  the  firet  term  of  a  geometrical  progression, 
and  the  amount  of  ^l  or  £1  tor  one  year,  at  the  given 
rate  per  cent,  the  ratio. 

2.  Carry  on  the  scries  up  to  as  many  terfiis  ai  ^^  given 
number  of  years,  and  tind  its  su.t 

5.  Multiply  the  sum  thus  found,  by  the  given  annuity, 
and  the  product  will  be  the  auiount  sought. 

EXAMTLES. 

1.  If  125  dols.  yearly  v(t\\t.  iw  annuity,  be  forborne,  (or 
unpaid)  4  years 5  what  Vvill  it  amount  to,  at  6  per  cent, 
per  annum,  compound  interest? 

1+1,06+1. 1^36fl,l9i(3i(5==:4,3746l6  sum  of  tl^c 
icries.*— Then,  4,37'4()1.jx  125=^546,8^^7  the  amount 
sought. 

OR  EY  TABLE  I 

Multiply  the  Tabular  number  under  tlie  rate  and  op» 
posite  to  the  time,  by  tlie  annuity,  and  the  product  will  be 
the  amount  sought* 

*Ttie  sum  of  the  series  thus  fmincUis  the  amount  of 
IL  or  1  (loUar  annuity .  for  the  ^ivea  iime^  ivhichvuuj  le 
foiindinl\tblc  TL  ready  calculated. 

Iience,  either  the  amount  criireseiit  ivorth  of  (innuiH€$ 
Wif  be  readUv  fmmd  by  Tahkafor  that  mirj^iose, 
:'8 


^i>  ANi^UlTlES    Oil    i'EK5I0NS> 

2.  If  a  salary  of  60  dollars  per  annum  to  be  paid  year- 
ly, be  forborne  20  years,  at  G  per  cent,  compound  In- 
terest; what  is  the  amount  ? 

Under  6  per  cent,  and  opposite  20,  in  Table  II,  jou 
will  find, 

Tabular   number=36,78559 

60  Annuity. 

Ms,  g2207,lS540=S2207,  IScts.  5m.^ 

4  v^  Suppose  an  Annuity  of  lOOL  be  12  years  in  arrears, 
U  is  required  to  find  what  is  now  due,  compound  interest 
being  allowed  at  ^L  per  cent,  per  annum  ? 

.Ans.  £1591  14s.SfiMd.  (by  Table    II.) 

4.  What  will  a  pension  of  120L  per  annum,  payable 
yearly,  amount  to  m  3  years,  at  51,  per  cent,  compound 
interest?  dns,  £STS  6s. 

II.  To  find  the  present  wortli  of  Annuities  at  Compound 
Interest. 

RULE. 

Divide  the  annuity,  &c.  by  that  power  of  the  ratio  sig- 
nified bj  the  number  of  years,  ana  subtiact  the  quotient 
from  the  annuity:  This  remainder  being  divided  by  the 
ratio  less  1,  the  quotient  will  be  the  present  value  of  tlie 
Annuity  sought. 

EXA:.irLES. 

I.  What  ready  money  will  purcha  se  an  Annuity  of  50L 
to  continue  4  years,  at  5L  per  cent,  compound  interest  ? 

"^^I^P^^^^^^^l  ==1,215506)5 

From  .  50 

Subti-act        41,13515 


Divis.  1.05—15=05)8,86487 


1 


TA'  TAELE  II 

Under  5  poT  cent,  and  even  with  4  j'cnrs. 
We  have  S554:j9cl —.present  vvOith  of  \l.  im*  4  years. 
Multiply  by  :^U=:Anmnty. 

Ans.   £  irr529r;)0=present  worth  of  the  annuity. 

G.  What  Is  the  present  worth  of  an  annuity  of  GO  dels, 
per  annum,  to  continue  20  vears,  at  6  per  cent,  compound 
interest?  "  ^??2S.  §688  19^  cts\-f. 

0.  What  is  oOZ.  peraiiiuuo,  to  continue  7  years,  %vorth 
in  rcaJv  monev^  at  6  rjer  cent,  compound  interest  ? 

Ans,  £167  9s,  5d.+ 

m.  To  liiid  the  present  v.ortli  of  Annuities,  Leases,  5cc» 
taken  in  Reversion,  at  Compound  Interest  ? 

1.  Divide  the  Annuity  by  that  power  of  tiie  ratio  deno- 
tetl  by  the  time  of  its  continuance. 

S.  Subtract  the  quotient  from  the  Annuity:  Divide  the 
remainder  f.>y  the  ratio  less  1,  and  th&  quotient  will  be  the 
present  worth  to  commence  immediately. 

3.  Divide  this  quotient  by  that  power  of  the  ratio  deno- 
ted by  the  time  of  Reversion,  (or  the  time  to  come  before 
tJie  Annuity  commences)  and  the  quotient  will  be  the 
present  worth  of  the  Annuity  in  Reversion. 

EXAMPLES. 

1 .  Wiat  ready  money  will  purchase  an  Annuity  of  50^. 
pa*yable  yearly,  for  4  years  :  but  not  to  commence  till  two 
years,  at  5  per  cent.  ? 

4th  power  of  1,05=1,215506)50,00000(41,13513 
Subtract  the  quotient =4 1,1 35 13 

Divide  bv  l,05--l==:.t>5V^>.36487 
£d.  power  6r  i,05=l,1025Vl77',29r(l60,8i36=£l60 
I6s.  Si.  lay,  present  w'orth  of  tl^iC  Annuity  in  Reversion. 

OR  BY  TABLIi  III. 

Find  the  present  value  of  IL  at  the  liven  rate  for  tire 
sum  of  the  time  of  continuance,  and  time  in  reversion 
added  togetl)er  ?  from  vvliirh  value  subtract  the  present 
worth  of  iL  for  the  time  in  reversion,  and  multiply  the  re- 
mainder bv  the  AnnTiity ;  tlic  product  will  be  the  answer. 


SOS  AVNUiTlES    OR    PENSIONS. 

T)iiis  In  Example  I. 
Time  of  continuance,  4  years. 
Ditto  of  I  eversloii;        2  * 

The  sum.                  =6  years,  gives  5,075692 
Tiniein  reversion,    =h2  years,  1,859410 


Remaindar,      5,2 16282  x50 
dns,  £160,8141 
S.  \Vhat  is  tlie  present  worth  of  75L  yearly  rent,  which 
is  not  to  coir.iTienco  urJi!  10  years  lience,  and  then  to  con- 
tinue r  ve^;  t  6  per  cent.  ? 

Ms,  £2S3  15s.  Od. 
5.  Yf  hat  is  tb.e  present  worth  of  the  reversion  of  a 
lease  of  GO  dollars  per  annum,  to  continue  20  years, but 
not  to  comnience  till  the  end  of  8  years,  allowing  6  per 
cent,  to  ihQ  purchaser  ?  J?2S.  S-^^l  TScts.  Q-^^m, 

IV.    To  f.nd  the  present  woi-th  of  a  Freehold  Estate,  or 
an  Annuitv  to  continue  forever,  at  Compound  Interest. 

rule: 

As  tlie  rate  per  cent,  is  to  1C^(.  :  so  is  the  yearly  rent  to 
tlie  value  reqi-ired.       exajniples. 

1.  Wiiat  istiie  worth  of  a  l**reeho]d  Estate  of  40^.  p^r 
annum,  allowing  5  per  cei^t.  to  ihe  purchaser  ? 

As  £5  :  £100  .  :  £40  :  £800  Ms, 

2.  An  estate  brings  in  pearly  150/.  what  would  it  sell 
for,  allowing  the  pinxhaser 6  per  cent,  for  his  money? 

'       £2500 


»'ir&o». 


V.  To  find  the  pi^esent  worth  of  a  Freehold  Estate,  ki 
Reversion,  at  (\>mr««)und  Interest. 

1.  Find  tlie  present  va^ue  of  the  estite  (by  thefore^o- 
in<5  rule)  as  thoii>  ■<  it  w\v^  to  be  entered  on  immediately, 
and  divide  the  fiald  value  f  v  i}\kit  power  of  the  ratio  de- 
noted by.th.e  time  of  rcvcrMoii.,  and  the  quotient  will  be 
the  presCiU  v:  "^     :  ''  "  ■  -   '  te  Lr  Ueve'-^iun. 

1.  Suppo;^e  a  iiv^;  :•  A  ;  c:-*.ueor  40L  per  annum  to  com** 
mence  two  years  hence,  be  put  on  sale ;  what  is  its  value, 
til  owing  the  purchnr ' .  "'      •  r  cent,  t 


As  5  :  100  :  :  40  :  800=tprcsent  worth  if  entered  on 
unmediatelj. 

Then,  1,05"=  1,1 025)  800.00(725,62358=725/.  12s. 
5icf.r=prcsent  worth  of  £800  in  two  years  reversion,  •/iws. 

OR  RY  TABLE  III. 

Find  the  present  worth  of  the  annuity,  or  rent,  for  the 
time  of  reversion,  whicli  subtract  from  tlie  value  of  the 
immediate  possession,  and  yon  will  have  the  value  of  the 
estate  in  reversion. 

Thus  in  the  foregoing  example, 
l5859410=present  worth  of  1^.  for  2  years. 
40=annuity  or  rent. 

74,376400 ^present  worth  of  the  annuity  or  rent,  for 
[the  time  of  reversion. 
From  800,0000  =va1ue  of  immediate  possession. 
Take     74,S764=i)resent  worth  of  rent. 


/;  725,623 6  =£725  12.<?.  5  Ad.  .(?«<?. 

2.  Suppose  an  estate  of  90  dollars  per  annum,  to  com- 
mence 10  years  hence,  were  to  be  sold,  allowing  the  pur- 
chaser 6  per  cent. ;  what  is  it  worth  ? 

Jlns.  S8S7,  59cts,  2m. 

o.  Which  is  the  most  advantageous,  a  term  of  15  ;y^ears, 
in  an  estate  of  100/.  per  annum;  or  the  reversion  of  sueh 
an  estate  forever  after  the  said  15  years,  computing  at  the 
rate  of  5  per  cent,  per  annum,  compound  interest  ? 

•^ns.  The  first  term  of  15  years  is  better  than  the  re* 
version  forever  afterv/ards,  by  £75  18.s.  7UL 


A  COLLECTION  OF*  QL^ESTIONS  TO  EXKRCISE 
THE  FOREGOING  RULES. 

1.  I  demand  the  sum  of  1748^  added  to  itself? 

Jns,  3497. 

2.  What  is  the  difference  between  41  eagles,  and  4099 
^imes.^  Jns,  lOcfs. 

3.  What  number  is  that  wliich  being  multiplied  by  21^ 
Uie  product  will  be  1365  .^  Jtus,  65. 


210 


qUESilONS    rOU    EXERCISE. 


4.  What  number  is  that  which  being  divided  by  19,  the 
quotient  will  be  72  ?  dns,  1368. 

5.  What  number  is  that  which  being  multiplied  bj  15, 
the  product  will  be  j  ?  Jins,  ^^, 

f».  There  are  7  chests  cl  drawers,  in  each  of  which 
there  are  18  drawers,  and  in  each  of  tlicsc  there  are  six 
divisions,  in  each  of  wliich  is  16L  Cs.  8d.;  how  much 
money  is  there  in  t]:c  v  "    '    " 

7.  bought  Z^-  -^ 
I  sell  it  a  pipe  i 
rest  for  v/liat  i\\(^ 

8.  Just  IG  vii. 
For  90  (In: 
IIow  manv  ,,= 
Will  14  el><j;icy 

9.  A  certain  quanf^ 
weeks,  how  many  mi: 
remainder  9  weeks  ? 

10.  A  grocar  beir^ht  a\\  ei^ual  q^i 
tiwA  coflee.  kn-  r4(;  la^--  '■"  *  '■■^  '•'^'--  ' 
the  sugar,  cu  ct;.  per 
the  coilee  ;  ic^jiureu  Ihc  ^iuct:,vi..  v  in  v.^vij,. 

Jns.  89,9.1.0,  r^oz.  8|fZr. 

11.  Bought  cloth  at  SU  a  yard,  and  lost  25  per  cent. 
Low  was  it  sold  a  yard  ?  .Qns,  93|cf5. 

12.  The  third  part  of  an  ramy  was  killed,  tlie  fourth 
part  taken  ]}risoners,  anti  1000  lied  ;  how  ?nany  were  in 
this  army,  h::v  -     '      ^:'''--\     :-|  how  many  captives  ? 

•  army^  HOO  killed^  and 


Jins.  £1£348. 
18  for  4536  dollars  5  h»w  must 
uiv  my  own  use,  and  sell  the 
Jns,  St29/60cfs. 


?s.  MSijds,  Sqrs,  ^na. 

■■r.'.  >--»  ]-tst  963  sheep  7 

t  it  will  last  the 

--■??s.  214. 

sugar,  tea, 

per  lb.  for 

.  per  lb.  for 


13.  Thomas  sold  1;"' 
and  received  as  much  vnone 


iits  a  piece, 
ceived  for  a 


certain  number  of  w^atcr-niclloiis,  v/l-jch  lie  sold   at  2:> 
cents  a  piece  5  how  m^ic- 
how  many  me] Ions  luid  I* 

*^]ns.  Each  received  ''^ " 

1 4 .  S  ai  d  J  ol  1  n  1 0  1 ;  \ 
9^.  2s.  but  the  money  is  cvvc-ir*-:: 
purse;  I  demand  IlOvs'  murh  luorv 


:  .<*j  anu 

noney  are  wortli 
l:-)uvs  as  u\uch  as  the 
was  in  it  ? 

Jns,  rS  15s 


qiJESTIONS  FOR  EXERCISE.  211 

15.  A  jcmng  man  received  210/.  whicli  was  §  ©f  liig 
elder  brotlier's  portion  ;  now  three  times  the  elder  broth- 
er's portion  was  half  the  father's  estate ;  what  was  the 
value  of  the  estate  ?  Ms.  £1890. 

16.  A  hare  starts  40  yards  betore  a  greyhound,  and  is 
not  perceived  by  him  till  she  has  been  up  40  seconds ;  she 
scids  away  at  the  rate  of  ten  miles  an  hour,  and  the  dog^ 
on  view,  makes  after  her  at  the  rate  of  18  miles  an  hour : 
How  long  will  the  ci>urse  hold,  and  vvhat  space  will  be  ran 
over,  from  tiic  spot  wlicre  the^dog  started  ? 

Ms,  GOJjSec,  and  5S0yds,  space, 

17.  What  number  multiplied  by  57  will  produce  just 
what  1S4  multiplied  by  71  will  do  ?  Ms,  l6Gff 

18.  Tlicreare  two  numbers,  whose  product  is  1610,  the 
greater  is  given  46 ;  I  demand  tlie  sum  of  their  squares, 
viid  the  cube  of  tlieir  difference  ? 

Ms.  T}ie  sum  of  their  squares  is  3341 .     The  cube  of 
their  aiffirence  is  1S31. 

19.  Suppose  tiiere  is  a  mast  erected,  so  that  -J-  cf  its 
length  stands  in  the  ground,  1 2  feet  of  it  in  tlie  water,  and 
f  f«  its  lengtli  in  the  air,  or  above  v/ater:  I  demand  the 
whole  length  ?  '  Vhis,  216  feet. 

20.  What  difference  is  tliCre  between  the  inteVest  of 
50Q.L  ai  5  per  cent,  for  12  years,  and  tlie  discount  of  the 
same  sum,  at  the  same  rate,  and  for  lUe  same  time  ? 

y  Ms.  £112  10.?. 

-  21.  A  stationer  sold  quills  ai.  \U.  per  thousand,  by 
which  he  cleared  |  of  the  money,  but  growing  scarce, 
raised  them  to  IS*;,  tid.  per  tliousaifd  ;  what  miglit  he  clear 
per  cent,  by  the  kitter  price? 

^     '  .^^ns.  £96 '7s,  S^^d. 

22.  Three  persons  purchase  a  TVest-iladia  sloop,  to- 
wards i]\Q  payment  of  wliich  A  advanced  -f,  B  l,and  C 
14DZ.  How  much  paitl  A  and  I],  and  what  ];art.  of  the 
vessel  had  C  ?  . 

Ms.  Jlim'ul  £26r^3^,   B  £CG3-f.,  and   C's  pari  of 
the  vessel  ivas  ^-J-. 

23.  What  is  the  purchase  of  1200/.  bank  stocl:,  ;it  I03| 
percent.  .^  Ms.  £\9A:o  iOf. 

£4.  Eough.t  27  pieces  of  Nankeeri-  oujh  II  \  wri^.  ni 


5i?v'  QUESTIONS   rOR  EXERCISE. 

14s.  4^d.  a  piece,  which  were  sold  at  18d.  a  vard^  rCt 
quired  the  prime  cost,  what  it  sold  for,  and  the  gain. 

£.  s,    d. 

CFrimecosty    19    8     ii 
JinsA  Sold  for^       23    5     9 
(^  Gain^  S  IT  '  7J 

9.5.  Three  partners.  A,  B  and  C,  join  their  stock,  and 
buy  goods  to  i\\Q  amount  of  /j  1025,5  ;  of  which  A  put  in 
a  certain  sum;  B  put  in....!  know  not  how  mudi,  and  C 
the  rest  5  they  gained  at  tlic  rate  of  9AL  per  cent. :  A's 
part  of  th-  gain  is  h,  B^s  \,  and  C's  the  rest.  Required 
Qach  man^s  particular  stock.  £, 

CA^s  stock  teas  512,75 

Jns.  <  B's ^  205,1 

iC's 307,65 

S6.  What  is  that  iiunibcr  which  being  divided  by  J,  the 
quotient  will  be  9A  ?  Ans.  15|. 

S7    If  to  my  age  there  addled  be, 

One-half,  one-third,  and  three  tiirics  thiee, 
Six  score  and  ten  the  sura  will  be  : 
Wliat  is  mj  age-- pray  sitew  it  nie  ? 

Ans.  66, 

28.  A  gentleman  divided  liis  fortune  among  his  three 
sons,  giving  A  9/.-  as  often  as  B  5L  and  to  C  but  SL  as 
often  as  B  7L  and  yet  C's  dividend  was  25S4L  ;  what  did 
the  wiiole  estate  aincunt  to  ? 

£ns,  £19466  2$,  Sd. 

29.  xV  gentleman  left  Ids  son  a  fortune,  ^  of  which  he 
spent  in  three  months  5  I  of  the  remainder  lasted  him  10 
months  longer,  when  he  had  only  2524  dollars  left;  praj 
'^^hat  did  his  fatlier  bequeath  liim  ? 

.^ns.  S58o9,  HScts.-r 
SO.  In  an  orchard  of  fniit  trees,  §  of  them  bear  apples, 
i  pears,  |  plums,  40  of  them  peaches,  and  10  cherries  5 
how  many  trees  does  tlic  orciiard  contain  ?     ^^m.  600. 

31.  Hiere  is  a  certain  number,  which  being  divided  by 
7,  the  quotient  resulting  multiplied  by  3,  that  product 
divided  by  5,  from  iAm  quotient  20  being  subtracted,  and 
SO  added  to  the  remainder,  the  half  sum  shall  make  Or 
c-^n  vou  tell  mc  the  number  ?  ''?''=>.  '•'-=.1. 


HUERTIONS  FOR  EXERCISE.  ^3 

3Z.  What  part  of  25  is  |  of  an  unit  ? 

Ms.  :^\. 
SS.  If  A  can  do  apwcc  of  work  alone  in  10  days,  B  in 
SO  dajs,  C  in  40  days,  and  D  ^ct  all  four 

about  it  together,  in  what  tvno  <  it  ? 

Ju6,  5^  days. 

34.  A  farmer  being  asked  how  many  sliccp  he  had,  an- 
swered, that  he  had  them  in  five  fields,  ia  the  first  he  had 
f  of  his  flock,  in  the  second  ^,  in  the  third  |,  in  the  foiii-tli 
j^,  and  in  the  fifth  450 ;  how  many  had  he  ? 

Ms.  1200. 

S5.  A  and  B  together  can  build  a  boat  in  18  days,  and 
witli  tlie  assistance  of  C  they  can  do  it  in  11  days ;  in 
what  time  would  C  do  it  alone  ?  Jlns.  28^  days. 

36.  There  are  three  numbers,  23,  25,  and  42;  what  is 
the  difference  between  the  sum  of  the  squares  of  tlie  first 
and  kst,and  the  cube  of  the  middlemost  ? 

Ans.  13332. 

S7.  Part  1200  acres  of  land  among  A,  B,  and  C,  so 
that  B  may  have  100  more  than  A,  and  C  64  more  than 
B.  Ms.  Jl  312,  B  412,  C  476. 

38.  If  3  dozen  pairs  of  gloves  be  equal  in  value  to  2  pie 
aC8  of  hoUand,  3  pieces  of  holland  to  7  yards  of  satin,  6 
yards  of  satin  to  2  pieces  of  Flanders  lace,  and  3  pieces  of 
rlanders  lace  to  81  shillings;  how  many  dozen  pairs  of 
gloves  may  be  bought  for  28s.  ? 

Ans.  2  dozen  pairs, 

39.  A  lets  B  have  a  liogshead  of  sugar  of  18  cv/t.  worth 
5  dollars,  for  7  dollars  the  cwt.  ^  of  which  he  is  to  pay  in 
cash.  B  iiath  paper  v/orth  2  dollars  per  ream,  which  he 
gives  A  for  the  rest  of  his  sugar,  at  2 1  dollars  per  ream. — 
Which  gained  most  by  the  bargain  ^ 

Ms.  A  %  S19,  20cf5. 

40.  A  father  left  his  two  sons  (the  one  1 1  and  the  other 
16  years  old)  10000  dollars,  to  be  divided  so  that  each 
share,  being  put  to  interest  at  5  per  cent,  might  amount 
to  equal  sums  when  they  would  be  respectively  21  years 
ef  age.    Ueqjuired  the  shares  } 

Alls.  5454 j^j  and  4545-^^  dollars. 

41.  Bought  a  certain  quantitir  of  broadcloth  for  5SS^ 


^14  quESTic:.,^    i-oR    Exr.Ficjf,?:. 

5s.  and  if  tlic  numl ings  v.iiich  It  cost  per  yard 

were  added  to  the  nui.ibei  oi  vanis  bouglit,  tlie  sum  would 
be  386 1  1  demand  the  iiuir.bor  of  yards  boii^^ht,  and  at 
what  price  per  yard  ? 

Jluf^.  C>o5  yds.  at  SI 5.  permriL 
Solved  by  Problem  Yi.  page  183. 
42.  Two  partners,  Peter  and  John,  bougiit^oods  tothe 
amount  of  1000  dollars;  in  the  purchase  of  \vhici«,  Peter 
paid  more  than  John,  and  Joiin  paid....!  know  not  how 
much :  Thej  tiien  sold  their  goGiis  for  ready  money,  and 
thereby  gained  at  the  rate  of  200  per  cent,  on  the  prims 
cost :  they  divided  the  gain  between  thcrri  in  proportion 
to  tlie  purchase  raoney  that  each  paid  in  buying  thQ  goods ; 
and  Peter  says  to  John,  Mj  part  of  the  gain  is  really  a  i 
handsome  sum  of  money  ;  1  wish  I  had  as  many  such  sums^ 
as  your  part  contains  dolhirs,  I  should  then  have  ^960000.: 
I  demand  each  man's  particular  stock  in  purchasing  th$: 
goods. 

Arts.  Feter  paid  600  dullars,  and  John  paid  400. 

THE   FOLLOWING    QUESTIONS    ARE    PROFOSED    TO 
SURVi^YOKS. 

1.  "Required  to  lay  out  a  lot  of  land  in  form  of  a  Ion* 
square,  containing  3  acres,  2  roods,  and  £9  rods,  that  shall 
take  just  100  loih  of  wall  to  enclose,  or  fence  it  round  5 
pray  how  many  rods  in  lengtli,  and  how  many  wide,  r.iust 
said  lot  be? 

*?72s.  31  rods  in  lengthy  and  19  i7i  breadth. 
Solved  by  Problem  Yl.  page  183. 

2.  A  tract  of  land  is  to  belaid  out  in  form  of  an  equal 
square,  and  to  be  enclosed  with  a  post  ami  rail  fence  5 
rails  high ;  so  that  each  rod  of  fence  shall  contain  10  rails. 
How  large  must  this  noble  square  be  to  contain  just  as 
many  acres  as  there  are  rails  in  the  fence  that  encloses  it, 
so  that  every  rail  shall  fence  an  acre  ^ 

dns.  the  tract  of  land  is  20  miles  sqnarL\and 
contains  256000  acres. 
Thus,]    mile=320  rods:  then   320x320->-l60=640 
acres  :  and  320x4x10=12800  rails.     As  640  :  12800  :  : 
12800  :  256000  rails,  which  will  enclose  256000  acres  ==* 
2{)  miles  square. 


:*15 

AM 

APPENDIX, 

CONTAINING 

SHORT  RULES, 
FOR  CASTING  INTEREST  AND  REBATE  5 

TOGETHER   WITH   SOME 

USEFUL  RULES, 

OJl  eiHDIKG  THE   CONTENTS  OF   SUPERFICIES,  SOLlDJS^ 


SHORT  RULES, 

?0R  CASTING  INTEREST  AT  SIX  PER  CENT 

To  find  the  interest  of  any  sum  of  shillings  for  anj 
number  of  days  lesis  thaaa  month,  at  6  per  cent. 

RULE. 

1. Multiply  the  sluliings  of  the  principal  by  the  num- 
r  of  days,  and  tliat  product  by  2,  and  cut  off  three 
2;ures  to  the  ri^ht  hand,  and  all  at)()ve  three  figures  will 
5  the  interest  in  pence. 

2.  Multiply  th^  figures  cut  oiT  by  4,  still  striking  off 
ree  figures  to  the  right  ijand,  and  you  will  have  tHe 
rthings,  very  nearly. 

EXAMPLES. 

1.  Required  the  interest  of  oL  Ss.  for  '25  days. 

5,8=108X25X2=5,400,  and  400x4=1,600 

Ans.  5d,  l,6^rs. 

2.  What  IS  ftie  interest ot  21 L  Ss.  for  29  days? 


210  APPENDIX/ 

FEDERAL  MONEY. 

II.  To  find  the  interest  of  any  number  of  cents  for  any 
number  of  days  less  than  a  month,  at  6  per  cent. 

RULE. 

Multiply  the  cents  by  the  number  of  days,  divide  tlie 
product  by  6,  and  point  oft*  two  figures  to  the  ridit,  and 
ail  the  figures  at  the  left  hand  of  i\\Q  dash,  will  be  tho 
Miterest  in  mills,  nearly. 

EXAMPLES. 

Required  the  interest  of  85  dollarSj  for  20  days. 
JS       cts,  mills. 

85=8500x20-~63=283,33  .te.  283  whicli  is 

£8cis.  3  milb. 
2.  What  is  the  interest  of  73  dollars  41  cents,  or  7341 
cents,  for  27  days,  at  6  per  cent.  ? 

Ans.  330  miilU^  or  ^3cts, 


III.  \yhen  the  principal  is^given  in  pounds,  shillings,  &c. 
New-England  currency,  to  find  the  interest  wt  any 
number  of  clays,  less  than  a  month,  in  Federal  Money, 

RULE. 

Multiply  the  Siullings  in  the  principal  by  ihQ  number 
of  days,  and  divide  the  ji-oduct  by  S6^  t\\e  quotient  witt 
be  the  interest  in  mills,  for  the  givei^  time,  nearly  5  omit* 
ting  fractions. 

EXA^IPLK. 

Required  the  iatert^t,  in  Federal  Money,  of  Q7L  15s- 
for  27  days,  at  6  per  cent. 

£  •    ^'      •'^• 
Aiis,  27    15=555 x27-7-56=416hu7Zs.=41c^s.  6?n. 


IV.  When  t:ic  principal  is  given  in  Federal  Money,  and 
you  want  the  interest  in  shillin2;s5  pence,  &c.  Ncw-Engj 
land  currency,  for  any  uum&er  of  days  Idss  than  a 


APPENDIX.  SIJJJ 

IIULK. 

Multiply  the  principal,  in  cents,  by  the  number  of  days, 
and  point  oflf  live  figures  to  tiic  ri«;ht  hand  of  the  product, 
whicn  will  give  the  interest  for  the  given  time,  in  shil- 
lings and  decin;a!s  of  a  shilling,  very  nearly. 

EXAMPLKS. 

A  note  for  65  dollars,  31  cents,  has  been  on  interest  25 
days;  bow  much  is  the  interest  thereof,  in  New-England 
currency  ? 

S  ct$,  s.  s.  (Lgrs, 

*5?2S.  65,Sl=65SJXi:5  =  l,63Sr5— I   T  >» 

Ret^iaukf. — In  tli*e  above,"  and  likewise  in  tiic  preced- 
ing practical  Rules,  (page  127)  the  interest  is  confined  at 
six  per  cent,  wliich  admits  of  a  variety  of  short  methods 
of  casting ;  and  when  the  rate  of  interest  is  7  per  cent,  av^ 
established  in  New-Yorl:,  &c.  you  may  first  cast  t!\e  in- 
terest at  6  per  cent*  and  add  tlieieto  one  sixth  of  itself, 
and  the  sura  vv^ill  be  the  interest  at  7  per  cent,  wh.ich  per- 
haps, many  times,  will  be  found  more  convenient  than  t^ie 
general  rule  of  casting  interest. 

EXAMPLE. 

Ucf[uired  the  interest  of  75L  hv  5  i-iontiis  :it  7  |)et 
cent.  5. 

7.5  for  1  month. 
'5 
-^    £.s.  a'. 
37,5=1  17  fi  lor  5  montiis  at  6  per  cent. 

4.1:=  6   3 

Jins   £2    3  9  for  ditto  at  7  per  cent. 


fjnOKr    METHOD    FOR     FINDING  THE  REBATE    Oi"   AN*-" 
GIVEN   SUM,   FOP.    MONTHS    AND    DAYS. 

^iULE. 

Dmijiii^li  tlie  inte'-estof  the  given  sum  for  the  time  by 
;  own  interest,  and  this  gives  the  Rebate  very  nearly- 

EXAMPLES. 

• .  What  is  the.  vr.Ki'xe  of  5Q  d.oUar*  for  sir  moiiU*^!.  'it 


S  cts. 

'he  intci-c-i  inonliiS^  is  1     50 


rjM 


cent  ? 


."^jzs.  Ilphaie.  gJ     45 
rebate  of  150L  for  7  montlis,  at  5  per 


/;• 

<:. 

fL 

** 

?■ 

G 

o 

Gi 

Interest  oi  1501.  ibr  T  ni:.)nllis,  is 
Interest  of  4Z.  7-.  C;L  lurr  i^ioiith 

.^iis,  £4    4  11^  nearlv. 
Bj  llic  above  Rule,  those  \\\\q  use  inte-est  tables  m 
tiieir  couutiiig-housesj  have  oulj  to  deduct  the  interest  of 
the  intcrcs^t,  and  the  remainder  is  ii\o  discount. 


A  concise  Rale  to  reduce  the  currencies  of  the  different 
States,  where  a  dollar  is  an  even  number  of  shillings^ 
to  Federal  Money, 

RULE  L 
Brliv^  (IjC  ^ivcn  sum  into  a  decimal  expression  by  in 
spectioii,  (;is  in  Problem  L  pv^ge  87)  then  divide  the  ^  z  :■  ^ 
by  ,S  in  New-England  and  by  ,4  in  Ncv/-York  curr^  r    , 
and  the  quotient  will  be  dollars,  ceutB,  &c. 


EXAMPLES 

L   8-3. 
Federal  iVIonev. 


1.  Reduce   '6Al.   Ss.  oAd.  New-England  currency,  to 


,3)54,415  decimally  expressed. 

Am.  SlSl.SScf.^. 
g.  Reduce  7s.  Hid.  New-England  currency, to  Fedc- 
i*al  Money. 

rs.  n|d==.<:0,599  then.  ,S),599 

*     Ms.  SI  ,33 
S.  Reduce  515L  16s.  lOd.  New-York,  &c.  currciicj^ 
to  Federal  •Money. 

j4)  5 135842  decimal 


nn<:  §1284,60* 


219 

4.  Reduce  19s.  j^il.  New-Ywk,  Sec.  currency,  to  Fede- 
ral money. 

,4)0,974  decimal  of  19s.  5Jd. 

S2.43i  Ans. 

5.  Reduce  64 ^  New-Kngland  currency,  to  Fadevai 
Money. 

,3)64000  decimal  expression 


S^l  5.35-1  Ans, 

Note. — By  the  foregoing  rule  you  may  carry  on  the 
decimal  to  any  degree  of  exactness :  but  in  ordinary  prac- 
tice, the  followin,-?;  Cordractlon  uuiy  be  useful. 

RULE  II. 

To  the  sisillings  contained  in  the  given  sum,  ann«x  -q 
times  the  given  pence,  increasing  the  product  by  2 ;  then 
divide  the  whole  by  tlie  number  of  shillings  contained  in 
a  dollar,  and  tlic  qv.otient  will  be  cents. 

EXAMPLES. 

1.  Reduce  45s.  6d.  New-England  currency,  to  Feto 
ral  Money. 

Gx8-f2  =  50  to  be  annexed. 
6)45j50     or     6)4550 

S7',58|  Jlns.       75S  cents,  =7,58; 

2.  Reduce  SZ.  10s.  Od.  New-York,  Sec.  currency,  t 
Federal  Money. 

9xS+2=r4  to  be  annexed. 
Then  8)5074  Or  thus,  8)50,74 

S  els.  

Ans.      634  cents.=^6  54  86,34  dns» 

N.  B.  When  there  are  no  pence  in  the  given  sum,  you 
must  annex  two  cyphers  to  the  shillings  ;  then  divide  as 
before,  &c. 

S.  Reduce  Si.  5s.  New-England  currency,  to  Federal 
Money 

S^.  5s.=65s.    Then  6)6501) 


renfs^     insf^  ,^n» 


/a 


SECTlOh 

Tlic  miperiicles  or  v.v\':i  1^5  anj  pame  suviace,  is  com- 
posed or  mader.p  of  H^KUires*  either  greater  or  less,»  iic- 
f>rdln^  to  the  dliien?!it  uieasiues  by  wiacii  the  dh^jen- 
;.ans  of  tlie  figure  ;ire  taken  or  ineaj-ii-red  : — a,nd  because 
i^iiiches  in  i^^igth  make  1  foot  of  long  measure,  there- 
fore, 12xl2™144.  u\Q  square  iiiches  in  a  superticiai  foot, 
vkc. 

Aht.  I.     To  find  tUe  rrrca  of  a  s^iuare  'lavin'^  equal 


Midtiplj  t]\Q  side  of  1:1-l2  square  into  it^^eii,  and  th2  pro- 
duct will  be  the  area,  or  content. 

EXATvfPLKS. 

1.  How  manj  sqnnre  feet  of  bu  ined  in 
the  floor  of  a  rooia  whicli  is  20  iect  square  ? 

S.  Suppose  a  S(Mi-  '  rods  o^i 

^\adi  side,  how  luati  v 
l^crE. — 160  squ:: 

Thereio^-p>  rC-xf'  :^r;----lG0™4«. 

.  .^ircr. 
Aa"\  i::.     ...  .square. 

iMultiply  tlic  ieiigiii  by  the  bread tii,  and  iae  product 
vnll  batfiearea  or  superficial  content. 

KXAMPJ.Kn. 

i.  A  certain  garden,  m  unra  oi  a  ion^|  square,  is  96  ft* 
long,  and  54  wide  ;  liow  uiaiiy  square  teet  of  ground  art* 
contained  in  it  ?  Jlns.  9G  x  54  ==5 1  ?,  4  square  feet 

2.  A  lot  of  land,  in  form  of  a  lon'^;  scnuirc,  is  120  rod? 
in  length,  and  GO  reds  wide  ;  how  v;ia:iv  acres  are  in  it  ? 

1^6x^0^7200  sq.  rods,  then,  y^s^4o  acreSy  Jns. 
5.  If  a  board  or  plank  be  £1  feet  long,  and  18  inches 
^road  5  how  many  square  f\ict  are  cgntained  in  it? 

IS  i}'ic^''St-=1^^   ■'^-  '    -<^ -7, 121  XT, 5  —  :^  1,5  Jins* 


Or,  in  measuring  boarilsj  jz^z  :Br.T  multiply  iiie  length 
iu  feet  by  the  breadth  ,n  iuclies/f.riil  divicic  by  12,  the 
quotient  v.ill  give  the  iinswer  in  square  teet.&c. 

Thus,  iu  the  tbiTgoiug  example,  21x18-7-12=31,5  as 
before. 

4.  If  a  board  be  ft  iiiclies  v/idc,  how  much  in  lengtk 
\\\\\  make  a  sciuure  foot  ? 

Rule.— Divide  144  by  the  breadth,  thus,    8)144 

Ans.  18  in. 

5.  If  a  piece  (f  lar.d  be  5  rods  wide,  how  many  rods  in 
length  will  wvdhi  an  aci-e  ? 

Rule. — Divide  1  GO  by  tlic  breadth,  and  the  quatiei^ 
will  be  the  length  required,  thus,  j)lCO 

Ans,     S2  TOih  in  length* 

Aht.  3.    To  measure  a  Triangle. 

.Definition, — A  Triangle  is  any  three  eernered  figui^e 
wliichis  bounded  by  three  right  lines.* 

RULE. 

Multiply  the  base  of  tlic  given  triangle  into  half  its 
pci-pendicular  height,  or  lialf  the  base  into  the  v/hole  per- 
pendicular, and  i\ie  product  will  be  the  area. 

EXAMPLES. 

1.  Required  tlie  area  of  a  triangle  whose  base  or  long- 
est side  is  S2  inches,  and  tlie  perpendicular  height  14 
iiichcs.  3^2x7=224  square  inches,   the  Jlnsiver. 

'■2,  There  is  a  triangular  or  tliree  cornered  lot  of  land 
whose  base  or  longest  side  is  51i»rods  ;  tlie  perpendicular 
from  the  corner  opposite  tlie  basi:;,  measures  44  rods :  how 
many  acres  doth  it  contain  ^ 

51,5x22^=1133  square  r()ds,=:x7  acres,  13  rods, 

'^Jl  Triangle  Qiiaji  be  either  right  anzled  or  oblique;  in 
e'iher  case  the  teacher  can  easily  give  the  scholar  a  right 
idea  of  th&  base  and  jperpendiculai ,  byviarkm^  it  down 
''  '.•  slate »  vaper^  Sec 


^  »2ii  AVPENDiX. 

TO  MEASURE  A  CIRCLE. 

Art.  4,    Tlie  diameter  of  a  Circle  being  given,  to 
find  the  Circu inference. 

RULE. 
As  r  :  is  to  22  :  :  so  is  tlie  given  diarr.ctcr  :  to  the 
circumference.     Or,  more  exactly,  Ab  113  :  h  to  355  *  : 
&ic.  file  diameter  is  found  inversely. 
^  Note. — T)\e  diameter  is  a  right  line  dra^rn  across  the 
circle  throug'n  its  centre. 

L  Wliatis  tlie  circumference  of  a  wheel  Vvliose  diam- 
eter is  4  fleet?— As  T  :  PS.  :  :  4  :  1^,57  t-:^  r^rciimfe 
rence. 

2.  What  is  the  circumference  cf  a  circ'  : -ame- 

ter  is  S5?— -As  7  :  £3  :  :  55  ;  liOJns\  :  sely 

as  £2  :  7  ::  no  :  S5,  the  dianr^^---  '" 

Art.  5.     To  find  the  : 
RULE. 
Multiply  half  the  diameter  by  half  the  circumference, 
and  the  product  is  ilie  area ;  or  if  tlie  diameter  is  givea 
without  the  circumference,  multiply  the  square  of  the 
diametei'  by  57854  and  the  product  will  be  the  iivea. 

EXA.MPLKS. 

1.  Requirel  the  area  of  a  circle  v.hosc  diameter  is  12 
inches,  anil    "  rence  37,7  inches. 

35  =half  the  circu inference. 
G==half  the  diameter. 


113,10  area  in  square  inches. 
2.  Kequired  the  area  of  a  circular  garden  whose  diame- 
ter i>s  11  rods  ^  .7854 
By  t:ie  second  metliDd,  11x11  ^  l^i 

Jns,  95,0334  rods, 

SI^CTION  2.    OF  SOLIDS. 

Solids  are  estimated  by  the  solid  inch,  solid  foot,  ^r, 
1728  of  these  inches,  that  is  12x12x12  make  I  c»b' 
or  golid  foot. 


APJ'ENDIX.  2i^3 

Art.  G.    To  measure  a  Cube. 

Definition, — A  cube  is  a  solid  of  sk  equal  sides,  each 
of  which  is  an  exact  square. 

RULE. 

Multiply  the  side  by  itself,  and  that  product  by  the 
same  side,  and  this  last  product  will  be  the  solid  content 
of  the  cube. 

EXAMPLES. 

1.  The  side  of  a  cubic  block  being  IS  inches,  or  f  foot 
and  6  inches,  how  many  solid  inches  doth  it  contain  r 

JL  in.   ft. 
1  6=1,0  and  1,5x1.5x1,5=3,375  solid  feet,  JIns. 
Orj  18xl8xlSr=5852  Sulid  inches,  and  -ff|i  =3,375. 

2.  Suppose  a  cellar  to  be  dug  that  shall  contain  12  fcQt 
every  way,  in  length,  breadth  and  deptli ;  how  many  solid 
feet  of  earth  mu&t  be  taken  out  to  complete  the  same  ? 

12x12x12=1728  solid  feet,  the  Jlnsiim\ 
Aut.  7.    To  find  trie  content  of  any  regular  solid  of  three 
dimensions,  length,  breadth  and  tliickncss,  as  a  ])iece  of 
timber  squared,  whose  length  is  mure  tluin  the  breadth 
and  depth. 

RULE. 

Multiply  the  breadth  by  the  depth  or  tliickncss  and 
tliat  product  by  the  length,  which  gives  i\\Q^  sc)!5d  co2itciit. 

EXAMPLES. 

1.  A  squaoe  piece  of  timber,  bein.*;  1  ^'Mt  6  inches,  m 
18  inches  broad',  9  inches  thick,  and  9  fe^t  ur  103  inches 
long;  how  many  solid  feet  doth  it  co:it:ibi  r 

1  h.   6  in. =1,5    foot. 

9  inches    =  ,75  foot. 

Prod.    1,125x9=10,125  s^^/a:    :.      ,.      ;. 
in,  in.  In.    solid  in. 
Or,  18x9xl08-=17496-r-1723=10J25  feet. 
But,  in  measuring  timber,  you  may  mul  tiply  the  bread  t!i 
in  inches,  and  tlie  depth  in  inches,  and  that  product  by 
the  length  in  feet,  and  divide  the  last  product  by  I-W, 
/ '  ..h  v.' ill  give  tte  solicfccontcnt  in  fcctp  &c. 


A^ 


'£24 


Arr-F.NDii:. 


2.  A  j>iect:  of  timber  being  16  iaches  broad,  11  iftchec 
thick,  and  20  feet  long,  to  fiiit^  the  cmiterut  r 
Breadth  16  inches. 
Depth      11 

Pi-od.     ir6x£0=3520    then,  3:£0 -v- 144 =24,4 /eef, 

the  Answer, 
r>.  Apiece  of  timber  15  inclics  broad,  8  inches  thick, 
and  25  [eat  long;  liow  many  solid  ^i^ct  doth  it  contain  ? 

fins.  SlO.S-^feet. 

Airr.  S,  V/iien  tlic  Isrciidth  and  thickness  of  a  piece  of 
timber  are  given  in  inches,  to  find  how  miicli  in  length 
will  make  a  solid  foot. 

Divide  ir£8  bj  tlie  product  of  the  bretidih  aiii!  depth, 
aiid  tiie  quotient  will  be  tlie  iengtli  niuking  a  suli<l  foot. 

KXAMPLKS. 

1.  ir  a  piece  of  timber  !}c  1 1  Inclics  broad  and  8  inches 
deep,  liow  n:any  incites  in  lengtli  will  make  a  solid  foot? 
Ilx8=:8S)ir28(19,6  inches,  Jlns. 

S.  If  apiece  of  timber  be  18  inches  broad  and  14  in- 
ches deep,  how  many  inches  in  length  will  make  a  solid 
foot  ? 

18xl4==.£5£  drjisor,  then  252)1728(6,8  inches,  Ans, 

Art.  9.    To  measure  a  Cylinder. 
iJ^/Ijzi^io/?.— A  Cylinder  is  around  body  whose  bases 
arc  circles,  like  a  round  column  or  stick  of  timber,  of 

equal  bigness  iioiu  end  to  end. 

RULE. 

Multloly  tj:e  sciuare  of  the  diameter  of  the.  end  by 
,7854  v/nich  gives  the  area  of  the  base ;  then  multiply 
the  area  of  ihn  u;!se  by  the  length,  and  the  product  v.  ill 
be  the  solid  coritcnt. 

EXAMPLE. 

What  is  the  solid  content  of  a  round  stick  oi  timber  cf 
equal  bigness  from  end  to  end,  whose  diameter  is  18  ii.- 
cbcs,  and  length  20  feet  ? 


18  111. =1,5  Tt. 

'.[uarc  2,25 X .ro54 ~  1^7^,715    area  of  the  base. 
X20  lrjm-t.h. 


.^i;:i^.  35jS-1300  solid  content. 
,    IS  inclios. 
18  inches. 

324  X  ^rs  ji  =:--r.a  J4.4G9G    iuclies;  area  of  the  base. 
CO  Ien<rth  in  leet. 


1  M)::0:^:.-,^!J20(S5,S43  ndld  fc(^t.  ^'his. 
T.  10.     To  uiui  iiov/  111  any  solid  teet  a  round  stick  (>f 
imber,  equally  thick  from  end  to  end,  will  contain 
,  hen  hewn  square. 

RULK. 
lultiply  twice  t':e  square  of  its  ftemi-dlameter  in  in- 
s  by  the  lenglU  in  feet,  tii-en  divide  the  product  by  144, 
i  the  quotient  v/ill  be  the  answer. 

EXAMPLE. 

r  the  diameter  of  a  round  stick  of  timber  be  22  inches 

.  its  length.  20  feet, ';  —  -r-.v  solid  feet  will  it  contain 
n  hewn  scpviare  ? 

;  i  Xllx2x2G--  '  :  f-t,  the  solidity  when 

.  n  square. 

V.  11.  Toiind  no^v  ujaijv  u-ct  oi' square  edged  bonrds 
Ta  given  thickness,  can  be  sav/n  froja  a  log  of  a  given 
'iameter. 

'ind  the  solid  content  (jf  V.n^  lo^.  when  niade  square, 
the  last  article — Then  r-^ay.  As  -vwq  thicknefs  of  Uiq 
ni  including  the  s:i\r  calf  :  is  to  the  solid  feet  :  :  so  is 
Inches)  to  the  number  of  feet  of  boards. 

low  many  feet  of  square  edged  boards-  1^  inck  thick, 
lading  the  «avv  c;tif,  can  be  sawn  from  a  log  20  fiet 
5  and  24  inches  diameter  ? 

12x12x2x20—144=40 /'f^  solid  content. 

As  U  :  40  :  :  12  :  384  feet,  the  Jlns, 


Art.  1^2.     Tiic  length,  breaiUh  and  deptli  fiTany  s^pare^ 
box  being  given,  to  lind  how  many  bushels  it  wilfcontain. 

RULE.  ^ 
Multiply  ihe.  lengtli  bj  the  breadth,  and  th.at  product^ 
by  tlie  deptli,  divide  the  last  product  bj  2150,425  the 
solid  inciieft  in  a  statute  bushel,  and  the  quotient  will  be 
ike  answer. 

KX  AMPLE. 

There  is  a  square  box,  the  length  of  its  bottom  is  50 
inches,  breadth  of  ditto  40  inches,  and  its  de])th  is  60 
inches ;  how  many  bushels  of  corn  v/ili  it  liold  ? 

50x40x60~^£150,425=:55,84+  or  55  bushels,  ihres 
pecks.  Arts. 

Art.  \^,  The  dimensions  of  the  walls  of  a  brick  build- 
ing being  given,  to  find  how  many  bricks  are  neces- 
sary to  build  it. 

HULE. 
From  the  whole  circumference  of  i>;0  vvall  measured 
round  on  the  outside,  STibtract  four  ti::r  m  Its  thickness, 
then  multiply  the  remainder  by  tlie  he;^:;-:,  and  that  pro- 
duct by  the  thickness  of  tlic  v/all,  pve3  u^Q,  solid  content 
of  the  whole  Vv  all ;  which  multiplied  by  the  number  of 
bricks  contained  in  a  solid  foot,  ci'v  es  tlic  ansvvcr. 


How  map.y  l- 
2i  inches  thick 
40  io-^t  wide,  r 
loot  thick  ? 

8x4x^,5-=.,  : 
=»21,6  bricks  in 

44+40-f^4-f-' 

a  so 

id:-       '      ^     "^ 

1 C:- 

Multiply 

^J 

1  ;J4  rc:y.:ii :i 
20  height. 

ix^s  wide,  and 
L'  44  feet  long,. 
.ills  to  be  one 

ilea  172»8-~80 

of  wall. 

•■.kncss. 


S280  solid  i^^itt  in  fiie  wha.e  walk 
Multiply  by     21,6  bricks  in  a  solid  foot. 

Fmdii  ct,    70848  bricks.  Ms. 


APPENDIX. 


•a^y 


1 


Art.  14.    To  find  t!io  tonnage  of  a  slirp. 
UULK. 

Multiplj  trie  -cngtii  of  the  keel  by  tiie  breadth  of  tne 
beam,  and  that  pro,  uct  by  the  depth  of  tisc  hold,  and  di- 
vide the  last  piodi.ict  t>v  93,  and  the  ciuoticrit  is  tlie  ton- 
nage. 

iCXAMiv:.:-".. 
Suppose  a  s]<ii>  72  feet  by  ttse  kcul,  aiid  ^24  feet  by  tlie 
beam,  and  12  feet  dcf^p ;  v.  Vat  is  the  tonnage  ? 

n2x24xJ2-r-95=«218.S+ions.  Jlns. 
KULK  11. 
Multiply  iiie  len^^ih  of  the  keel  by  tlie  breadth  of  the 
beam,  and  that  product  by  half  the  bieadth  of  the  beam, 
and  divide  by  95. 

EXAMPLE. 

Asliip  84  feet  by  the  keel,  28  feet  by  the  beam;  Avhat 
is  tlie  tonnage  ? 

84x28x14-^95=350,29  tons.  Jris. 

Art.  15.    From  the   proof  of  any  cable,  to  find   the 
strength  of  another. 

RULE. 

The  strength  of  cables,  and  consequently  the»  weights 
of  tlieir  anchors,  are  as  tlm  cube  of  their  peripheries. 
Therefore;  As  the  cube  of  the  periplicry  of  any  cable, 

Is  to  the  welglit  of  its  anchor  : 

So  is  tiic  cube  of  tiie  per'n)]iery  of  any  other  cabU, 

To  the  v/eiglit  of  It?  anc!)or. 

KXAMPLKS. 

!•  If  a  cable  G  Inches  about,  re({Uire  an  anchor  of  2| 
cwt.  of  Avhat  v/cight  must  an  anclior  be  for  a  12  inch  cable  ? 

As  6x6x6  :  2icwL  :  :  12x12x12  :  IScwt.   Jlna. 

2.  If  a  12  inch  cable  require  an  anchor  of  18  cwt.  what 
wuist  the  circumference  of  a  cable  be,  for  an  anchor  of 
2i  cwt.  ? 

c:vt.  cwL  tn. 

As  13  :  12x12x12  :  :  2,25  :  216^216=6  dns. 

Art.  16.    Having  the  dimensions  of  two  similar  built 
s!up=>  of  a  diiTerent  i  opacity,  with  the  burthen  of  onft 

;?':  i:i%in^  to  frnd  thcburSien  of  lli€  oth*.ir , 


22ft  APPENDIX. 

RULE. 

Tlie  burtliens  of  similar  built  ships  are  to  each  oti:  r, 
as  the  cubes  cl  tl  2it  like  dimensions. 

EXAMPLE. 

If  a  ship  of  500  tons  burthen  be  T5  ieet  long  in  the  k*  ■  ', 
I  demand  the  burthen  of  another  ship,  whose  keel  is  '.'  'J 
feet  long  r         -  '         T.cwt.qrsJi'. 

As  75x75x75  :  300  :  :  100x100x100  :  711  2    0     i ■  '    - 


DUODECIMALS, 

CROSS  MULTIPLICATION, 
i.S  a  rule  m?Ae  use  of  by  \vorknien  and  artificers  in  c<:::*- 


inguptkc  contents  of  tlieir  wcrk. 

RULE. 

1.  Under  the  multiplicand  write  thQ  corresponding  dc  ■ 
uoininaticns  of  tiie  multiplier. 

2.  i\ii'h"::]v  each  ten:  '         "  ^^   "-and^beginnh^ 

(•.tthe  los\^v-t5  by  the  h=  :i  in  the  m:::    - 

piicr.  and    .\  rite  the  ]--■  its  respeci'^  ■ 

term?  ^^;-   rvmg  to  c  ;.,  v  12jfrom  e^.    . 

lower  denominatioii  (o  i 

5.  Ih  (;13  sn.n"ie  Tuni-;  '•  niulthilKT '   < 

by  the  i;':'-        '        •  :isTi^iii:itL;>n.  !-i  the  nmltiph-^ 

;:rid  s'^::  '  rvvwjnenhice  removed  to  i   ■' 

ri-hf-: 


niidtii)li'^i\ - 


ling  the  result  of  eaca  te: 
of  those  in  tiU!  mu!t:plicn 


Multiply     7 
By        "     4 


F, 

L 

F. 

X 

4 

6 

9 

7 

■3 

8 

9 

" 

09     0    ''  ..J     6  91  IC 


APPENDIX  £29 

F.   I.  F.   L  F.  L 

Multiply      4    7  3     8  9  7 

Bv  5  10  7    6  S  6 


ti 


Product, 

2G    8  10 

MuUipiv 
By 

F,    I. 

S  11 
9    J 

Product. 

36  10    7 

27    G  32    6     6 


F.    7. 

6  5 

7  6 


48     1     G 


TEET,  INCHES  AND  SECONDS. 
F.     L      " 

Multiply    ,9    8    6 
By  7    9    3 


F.   L 

7  10 

8  U 

M 

C9  10 

% 

[tiplier. 


67  11    6   '"      =prod.  by  the  feet  inthemul- 
7    3    4    6    ''"=ditto  by  the  inches. 
2    5    1    6=ditto  by  the  seconds. 


75    5    3    7    6  Ms. 


F.    L    "  F.    L    " 

Multiply     7    19  5    6    7 

By        "     7    8    9  8    9  iO 


Product,   55    2    9    3     9         48  11     2    8    10 


How  many  sqnare  ieet  in  a  board  16  feet  9  inches' 
lon^,  and  2  feet  3  inches  wide  ? 
Sy  Duodecinuds. 

F.    L 

\Q    9 


4    2 


Am.  57 


Ih, 

Decimals. 

h\ 

L 

15 

9^16,75  feet 

2 

3=  2,25 

8375 

3S50 

3350 

F. 

20 


Ms.  37,6875 «. 37     8    $ 


^0  APPr^NDIX. 

TO  mp:asure  loads  of  wood. 

i         ^  RULE. 

%       Multiply  the  length  by  the  bread tii,  and  tlie  product  by 

t,  th.e  depth  ©r  height,  which  will  give  the  content  in  soli(i 

feet  5  of  which  64  make  half  a  cord,  and  128  a  cord. 

EXAMPLE. 

How  many  solid  i^eet  are  contained  in  a  load  of  wood, 
7  feet  6  inches  long,  4  feet  2  inches  wide,  and  2  feet  3 
inches  high  .^ 

7  ft.  6  in.— 7,5  ami  4  fL  2  2;i.=4,l6r  and  2  ft  3  z?2=» 
2,25;  then,7,5x4,l6r='si52525x2,25==70,3181'S5so^if^ 
feetj  Jns, 

But  loads  of  wood  are  commonly  estimated  by  the  foot* 
allowing  the  load  to  be  8  feet  long,  4  feet  wide,  and  then 
£  feet  high  will  make  half  a  cord,  which  is  called  4  feet  of 
wood ;  but  if  the  breadth  of  tiie  load  be  less  than  4  feet, 
its  height  must  be  increased  so  as  to  make  half  a  cord 5 
which  is  still  called  4  feet  of  wood. 

By  measuring  the  breadth  and  lieighthof  the  load,  the 
content  may  be  found  by  the  following 
RULE. 

^lultiply  the  breadth  by  the  height,  and  half  the  pro- 
Aict  will  be  the  content  in  feet  and  inches. 

EXAMPLE. 

Required  the  content  of  a  load  of  wood  which  is  3  l^et 
9  inches  wide  and  2  feet  6  inches  high. 
By  Duodecimals.    By  Deciv.ids. 
F.hu  F. 


S     9 
S    6 

7    6 
1  10 

6 

Vu73 

1875 
750 

9    4 

4    8 

6 
5 

<),575 

F,  in. 

4,6875=4     8i, 

-^rtS.4    8    5  4,6875=4     8^,  or  half  a  cord  and 

S}  inches  over, 
^he  foregoing  method  is  concise  and  easy  to  those  who  are  well 
'-4quajnted  w'lih  Duodeciiiials,  but  the  following  Table  will  give  th« 
©Oirtent  of  any  lc^ad  of  wood,  by  inspection  only,  sufficieDtly  exaiQf 
Ibi'  corasoD  practice ;  wbicb  nui  Iw  fo^d  very  ccryeaieat. 


*'?! 


.3  T.ii 

V; 

'(lulih^ 

Ilei 

-;«<, 

and  Content. 

BreaAth.] 

J'L  in. 
2    6 

Kei'^AJ  in/cf  ?.| 

7/ic//M, 

1 
15 

i^" 

I: 

T 

2    3| 

4| 

5      G|7|8|9|10l 

11 

30 

s 

T 

IT 

T 

5 

"cr 

7     9 

10  11112 

14 

T 

16 

31 

4r 

.e 

I 

5 

4 

5 

G 

8     9 

10 

12  13 

14 

8 

|1G 

32 

48 

6^ 

1 

3 

4 

5 

7 

8     9 

11 

12  13 

15, 

!         9 

17 

33 

49 

66 

1 

3 

4 

G 

-' 

8 

9 

11 

12  14 

14 

10 

lir 

34 

^>l 

GS' 

2 

3 

4 

G 

7 

9 

10 

11 

43  14 

ir; 

,       11 

118 

35 

53 

70 

o 

3 

4 

6 

r 

9 

10 

12 

1SJ15|1GJ 

IT  0 

18 

SG/J4 

72 

¥ 

3    J 

^  1 

O 

9   11 

12  14  15  17' 

!     1 

19 

371.56 

74 

o 

3    5 

G  { 

8 

9   11 

12  14  16  17 

o 

19 

SS  -57 

76 

o 

3i5 

'6  ! 

8     lOJll 

13  14  16  i:^ 

1         3 

jl9 

39  59 

78 

2 

3 

5 

«       1 

8     10  111 

13  15  16  18, 

4 

20 

40  60 

8C 

2 

3 

5 

8     10 

12 

13  15  17  18 

5 

21 
2i 

41  62 

42  33 

82 
S-1 

2 

3 

5 

7 

8     10 

12 

14  16  17  19 

6 

"o" 

4 

5 

_,^_. 

9  ill 

12|14|16|1S 

19 

7 

o:^ 

43  G4 

SCI 

2!4 

5 

i 

9    11 

13 

14  16  18 

20 

8 

iS^ 

44  oG 

88  1  2  U 

6 

7 

9    11 

15 

15  17  18 

2Q 

)         9 

!.>; 

45^68 

90    2,4 

6 

9 

11 

13 

15  17  19 

2f 

10 

ii-jS 

46  69 

92'  2    4 

6 

7 

9 

12 

13 

15  17  19i2l| 

!     ^^ 

1^23 

4770 

94i  2    4 

6 

8 

10 

12 

14 

16  18  20 

22 

4     0  i 

[24 

48  7.2 

961  2  14 

6 

8 

10 

12 

14 

16  18  20 

22 

TO  USE  THE     FOREGOING  TABLE. 

First  measure  the  breadth  and  height  of  your  load  to  the 
nearest  average  inch  ;  then  find  the  breadth  in  the  left  hand 
column  of  the  table  ;  thfn  move  to  the  right  on  the  same  line 
til!  you  come  under  the  height  in  (eM,  and  you  will  have  the 
content  in  inches,  answerinf^  the  fcetj  to  which  add  the  cor.- 
tent  of  the  inches  on  the  right  and  divide  the  sum  by  12,  and 
you  will  have  the  tvue  content  of  the  load  in  feet  and  inches. 

Note.— The  contents  answering  the  inches  being  always 
small,  may  be  added  by  inspection. 

EXAMPLES. 

1.  Aduiit  a  loa4  of  wood  is  3  feet  4  inches  Vvide,  and  2  feet 
10  inches  high  ;  required  the  content. — 

Thus,  agaiiist  3  ft.  4  inches,  and  under  2  feet,  stands  40  inch- 
es ;  and  under  10  inches  at  top,  stands  17  inches:  then  40-f- 
17=57  true  content  in  inches,  which  divide  by  12  gives  4  feet 
9  inches,  the  answer. 

2.  The  breadth  being  3  feet,  and  height  2  feet  8  inches ; 
required  the  content. — 

Thu9,  with  bf2»4th  S  feet  0  inches,  and  under  2  ft^Jt 


S.'i 


Al'PKNDIX. 


atop,  stands  S6  iiul.cs ;  and  uwdcr  8  inches^  sta&da  1%^ 
inches :  now  35  and  12,  make  48,  the  answer  in  iacheai 
arid  48-7-12=4  feet  or  just  half  a  cord. 

3.  Admit  the  breadth  to  be  3  feet  11  inches,  and  height 
3  feet  9  inches ;  required  the  content. 

Under  3  fee.t  at  top,  stands  70 ;  and  under  9  inches,  is 
18  :  70  and  18,  make  88-~-12~7  feet  4  inches,  or  7  ft.  1 
qr.  2  inches,  the  answer* 


TASLE  T. 

Showing: the anumnf  of  £'i^ or  %\,at  5 and  6 pcT cent. per 

ammm^  Om^womid  Interest^  for  20  years. 


i'ra. 

5  2)er  ceut.\Gpii>r  cent. 

1  Vs.  15  per  cent. 

6  per  cent. 

1 

1,05000 

1,06000 

11 

1,71034 

1,89829 

2 

1J0250 

1,12360 

12 

1,79585 

2,01219 

S 

1,15762 

1,19101 

13 

1,88565 

£,13292 

4 

1,21550 

1,26247 

14 

1,97993 

2,26090 

5 

1,27628 

1.33822 

15 

2.07893 

2,39555 

6 

1,54009 

1,41851 

16 

2,18287 

g.54727 

4 

1,40710 

1,50565 

17 

2,29201 

2,69277 

t    8 

1.47745 

1.59S84 

18 

2,40661 

^  2,85433 

f    9 
10 

1,55132 

1,68947  i  19 

2,52695 

3,02559 

1,62889 

1,7<)084  1  20 

2,65329 

3,20713 

VII,    The  weights  of  the  coins  of  tlie  United  States. 
pwt.    gr. 
11      6  1 
5     15    V 

2   m  ' 

17 
8 
4 
1 


Ea^le^, 

llalf-Ea^Ies, 

Quarter-Eagles, 

Ooilars, 

llalf-Dollars, 

Quarter-Dollars, 

Dimes, 

Half-Dimes, 

Centr5, 

Half-Cents, 


16 

8 


Stuiidaid 
Gold. 


Standard 

.  f"  Silver. 

2ajJ 

8   K^PP^'- 


Tlie  standard  for  gold  coin  is  11  parts  piire  gold,  and  one  part  al» 
foy— the  alloy  to  consist  of  silver  and  copper.  The  sianuord  for 
silver  coin  is  U25  perts  doe  Uk  179  parU  a!lov~^.V,  alloy  to  be  wtiolU 
If  copper. 


APPEjiDiX. 


ANNUITIES 


Table  ii. 

Showing  the  amoiint  of 
£  1  annuity^  forborne 
Jor  31  years  or  under, 
at  5  and  6  jier  cent, 
comvound  interest. 


Frs. 


/ 

8 

9. 

10 


11 
12 
15 
14 
15 


16 
IT 
IS 
19 

20 


21 

23 

24 

25 

26" 

27 

28 

29 

SO 

SI 


l.OUOOOO 
2,050000 
3,152500 
4,310125 
5'525631 


6,801913 

8.142009 

9^549109 

11,026564 

12.577892 


^4,206/87 
15,917126 
17,712982 
19,598632 

21,578564 


23,657492 
25,840366 
28,132385 
50,539004 
33,065954 


1. 000000 
2,060000 
3,183800 
4,374616 
5,637193 


6,975319 

8.393838 

9,897468 

11,491316 

15,180770 


I4,97i6*ic? 
16,869942 
18.882133 
21,015066 

23,275969 


25,672528 
28,212380 
30,905653 
33,759992 
36,78559: 


35,719252 
38,505214 

41,430475 

44,501999 

47.727099) 

5],Tr3'45^ll59,156382 

54fiG0V2C>\QS,705T65 


39,992727 
43,392291 
46,995828 
50,815578 
54,854512 


58,402583 
62,322712 
66,438847 
70,760790 


68,528112 
73^639798 
79,058186 
84,801677 


TABLE  ill. 

Showing  the  jiresent 
ivorth  of£  1  annuity, 
to  continue  for  51 
years,  at  5  and  6  per 
cent,  compound  int. 


0,952381 
1,859410 
2,723248 
3,545950 
4.329477 


5.075692 

5,786278 
6,463213 
7,107822 
7,721735 


8,306414 
8^863252 
9.393573 
9,89864  f 
10,579658 


10,837769 
11,274066 
M  ,639587 
12,085321 
12,462210 


125821153 
13,163003 
13,488574 
15,798642 
14.093944 


0,943396 
1,853393 
2,673012 
3,465106 
4,212364 


4.9J7S24 
5,582381 
6,209794 
6,801692 
7.360087 


7,886875 
S,385844 
8,852685 
9,294984 i 
9,712249 


10,105895 
10,477260 
10,827603 
11,158116 
11,469921 


11,764077 
12,041532 
12,305330, 
12.550357 
12,783356 


14,375185  13,003160 


14,643034 
14,898127 


15,14^073^15,5907 
15,372451  """^ 
15,592810 


^5,210534 
15,406164 


>1 
'15,764851 
15,929086 


^  €34 


APPKNPXX. 

TABLES.^ 


X  TIE  three  following  Tables  are  calculated  agreeable 
to  an  Act  of  Congress  passed  in  November,  1 792,  making 
foreign  Gold  and  Silver  Coins  a  legal  tender  for  the  pay- 
ment of  all  debts  and  demands,  at  the  several  and  respec* 
tive  rates  following,  viz.  The  Gold  Coins  of  Great-Bri- 
tain and  Portugal,  of  their  present  standard,  at  the  rate  of 
100  cents  for  i:^\ery  2.7  grains  of  the  actual  v/eight  there- 
of.— Tiiose  of  France  and  Spain  27|  grains  of  the  actual 
weight  thcieof. — Spanish  milled  Dollars  v/eighing  If 
pwt.  7  gr.  equal  to  100  cents,  and  in  proportion  for  the 
parts  of  a  dollar. — Crowns  of  France,  weighing  18  pwt» 
ITgr.  equal  to  1 10  cents,  and  in  proportion  f  >r  the  parts 
of  a  Crown. — Tliej  have  enacted,  tliat  every  cent  shall 
contain  208  grains  of  copper,  and  every  half-cent  104 
grains. 


TABLE  lY. 

^Feighis  of  several  pieces  of  English^  Portuguese,  and 
French  Gold  Coins, 


Johannes 

FwL 

[    (Jr. 

Dols.  Cis.  M. 

18 
9 
5 
2 
5 
2 

16 
8 
4 
6 

6 
15 

6 
15 
12 

G 

S 

16       0     0 
8 

4     6(>| 
2     SS^ 

4  59      8 

£     29     9    . 

14     45      2   1 

r     22     6  1 

5  61      3  U 

6  14     si 

Single,  ditto, 

finglish  Guinea,  .... 

Half,        ditto, 

French  Giwnea,   .  .  .  .  , 

Half,       ditto, 

4  Pistoles, 

C  Pistoles, 

I  Pistole, 

Moidore, 

APPJINPIX. 


S55 


Ip5 

•0.7»i-«^-        ^>  -  t- CO        »:^  ^t- «  CS  10  0«0  W-        W«0»0 

-n 

3 

•• 

s 

1 

•^  CI  CO  -^  kC  CO  t-  CO  CV  O  ^  G)  CO  "«f  «0  «  r-  CO  {y>    M  -«  ©»  60 

1 

I 

CO  CO  CT> 

xo  CO      tc  6<  C5  o  G*  CO  U-;  -.  M  T*  ^  <•£  T*  o  i^  eo 

d 

ooir- *  CO  —  i?5  n  -?■»  T  o  CO  t-  —  •*  CO  e-»  «o  crs  «0  'X.  o  eo 

-«'-  —  G^G^&<cocO'r'!^TJ•^r;lOi00t^cp^-t-ooco 

—  e^eOTfio-ci-ccc-.o  —  S'icorp«5or-Gooo»-'«^r> 

I 


t-  to  o  -3*  CO  s-i  — <  r-  to  lo  >»•  CO  G<  -* 


iC:  -<  G!  CO  •*i«  '-O  '-0  C--  CO  C^3  ffi  O  • 


~ 

SlCO-^iCtOt^OCOO 

- 

ei  eo  rj"  o  CO  r^  CO  o 

- 

s* 

eo 

t-  -*  —  CO  o  e^  o  to  CO      f  T?  P-.  00  «.o  e<      <a3  ri*  —  &<  uo  «» 


?»-ieic^"^»o®l*c90>0'-4e*co'-*ir>ot-o»oor-c?co 


216 


APPENDIX. 


YIII.  TJiBLE  of  Cmts,  answering  to  the  Cufrmcks 
of  the  United  States,  with  Sterling ,  Sfc. 
Note. — The  figures  on  the  right  hand  of  the  space, 
show  the  parts  of  a  cent,  or  mills,  &c. 


6s.  to 

8s.  to 

7s  6d. 

4s.Sd. 

5s.  to 

4s.6d. 

4s.  Und. 

tlie 

the 

to  the 

to  the 

the 

to  thp 

to  the 

Boll 

Boll 

Boll. 

Boll. 

Boll. 

Boll. 

Bollar, 

P. 

cents. 

cents. 

cents. 

cents. 

cents. 

cents. 

cents. 

1 

1  3 

1  0 

1   1 

J  7 

1  6 

■  1  Sj  1 

7 

s 

2  7 

2  0 

2  2 

S  5 

3  3 

3  7 

3 

4 

5 

4  1 

5  1 

3  3 

5  5 

5 

5  5 

5 

1 

4 

5  5 

4  1 

4  4 

7  1 

6  3 

7  4 

6 

8 

5 

6  9 

5  2 

5  5 

8  9 

8  6 

9  2 

8 

5  ' 

6 

8  S 

6  2 

6  6 

10  7 

10 

?*  1 

10 

2  f 

7 

9  7 

7  2 

7  7 

12  5 

11  6 

1'19 

11 

9 

8 

11  1 

8  3 

8  8 

14  2 

13  3 

.^B 

13 

6 

9 

12  5 

9  3 

10 

16  ^ 

15 

16  6 

15 

3 

10 

13  8 

10  4 

ii  1 

17  8 

16  6 

18  5 

17 

11 

15  2 

11  4 

12  2 

19  6 

18  3 

20  3 

18 

S. 

1 

16  6 

12  5 

13  3 

21  4 

20 

£2  2 

20 

£ 

S3  3 

25 

26  G 

42  8 

40 

44  4 

4) 

•i 

5 

50 

57  5 

40 

64  2 

60 

66  6 

61 

5 

4 

66   6 

50 

53  3 

85  7 

80 

88  8 

82 

5 

83  S 

62  5 

66   6 

107  1 

100 

111  1 

102 

5 

6 

IGO 

75 

80 

123  5 

120 

133  S 

123 

7 

'116  6 

87  5 

93  3 

150 

140 

155  5 

143 

5 

8 

133  3 

100 

106  6  171  4 

160 

177  7 

164 

1 

9 

150 

112  5 

120 

192  8 

480 

200 

184 

5 

10 

166  6 

125 

133  3 

214  2 

200 

222  2 

205 

1 

11 

183  5 

137  5 

146  6 

235  7 

220 

244  4 

225 

6  ; 

12 

200 

150 

160 

257  1 

240 

266  6 

246 

1  < 

15 

216  6 

162  5 

173  S|278  5 

260 

288  8 

^66 

6  1 

14 

233  3 

175 

186  O'SOO 

280 

311  1 

287 

1  1 

15 

250 

187  5 

200 

32  i  4 

300 

333  5 

307 

6  i 

16 

266  6 

200 

213  3 

342  8 

320 

355  5 

328 

£"  i 

17 

283  3 

212  5 

226  6 

564  2 

340 

348 

/ 

18 

300 

225 

240 

385  6 

360 

400 

369 

o 

19 

316  6 

237  5 

253  3 

407  1 

380 

^i^r  ^ 

'■  '  ^  '"•; 

20 

333  3 

250 

266  6 

428  5 

:n') 

APPENDIX. 


isr 


TABLE  IX, 

Showing  the  value  of  Federal  Money  in  other  Currencies, 


Federal 
Money, 


J>rtW'En^ 

land^  Vir- 

^iniaj  ami 

Kmtucky 

currency. 


*MeW'Fork 

andJVorth- 

Carolina 

currency. 


^,  Jersey  y 
Fennsylva- 
nia^  i)ela- 
ware,  and 
Maryland 
currency. 


South-Car- 

olina 

t,  and 

Georgia 

currency. 

s. 

d.  ■ 

0 

OJ 

1 

1 

0 

n 

0 

'2k    - 

0 

^li 

r     0 

H 

0 

4 

0 

44 

0 

5 

0 

5J 

0 

6i 

0 

6* 

0 

7i 

0 

n 

0 

8i 

0 

9 

0 

94 

0 

10 

0 

10S 

">" 

!ia^•c 

Cerits, 
1 

S 

4 

5 

6 

7 

8 

9 
10 
11 
12 


3 

S4 

4i 
5 

5J 
64 

rj 

8 
8j 
9i 
10 

A  L.. 


1 

S 

S| 
41 
51 
6J 
7i 
8J 
94 
104 
114 
04 
14 
24 
Si 
4i 
5^ 


s. 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 

I 
1 
1 

1 
1 


1 

1* 

21 
34 
4^ 


9 

10 

lOJ 

113 

04 

14 

24 

H 

4i 


RNO Wail  men  bvl 
in  tor  .111(1 

received  to  my  full  satis^favini       5 J 
tiiis  (lay  01  in  the  year  of  oiir  ^. 

demised  and  to  farm  let,  and  "do  by  these  presents,  de- 
mise and  to  farm  letj  inito  t!iC  said  P.  V.  his  heirs,  execu- 
tors, admiTiistrators  and  asssi<;ns,onu  certain  piece  oFIand, 
lying  and  being  situated  in  said  bounded,  &g 

fHere  describe  th.e  boundaries"!  with  a  dwellin*^ -house 
tliereon  standing,  tor  the  term  ot  one  year  from  this  date. 
To  HAVE  and  to  hold  to  him  the  ^aid  I'.  V.  his  \w\vs, 
^xeoutorS}  administrators  and  assign*  foi  sail  tern?,  tor 


S3S 


APPEKOIX. 


A  FEW  USEFUL  FORMS    IX  TRANSACTING    IJUiilNKSS. 

AN  OBLIGATORY  BOND. 

KNOW  all  men  by  these  presents,  tluat  I,  C.  D.  of 
in  the  couiitj  of  aiu  held  and  iinnly  bound  to 

H.  W.  of  in  the  pen-il  sum  of  to  be  paid 

II.  W.  his  certain  atternej,  executors  and  administrators; 
to  which  payment,  well  and  truly  to  be  inade  and  done, 
I  bind  myself,  my  heirs,  execiiturs  aiHl  administrators, 
firmly  by  these  presents.  Signed  v.  ith  my  hand,  and 
sealed  with  my  seal.     Dated  at  fms  day 

of  A.  D. 

The  condition  of  this  obligailo,  .  That  if  the 

above  bounden  C.  D.  &c.  [Here  iu^ert  the  condition,'] 
Then  this  obligation  to  be  void  and  of  none  eftect ;  other- 
wise to  remain  in  full  force  and  virtue. 
Signed  J  sealed  and  delivered  J 
in  the  presence  of         ^ 


A  BILL  OF  SALE. 

KNOW  all  men  by  these  presents,  tiiat  I,  B.  A.  of 
for  and  in  consideration  of  to  me  in  hand  paid  by 

D.  C.  of  the  receipt  whereof  I    '    ' 

knowledge,  have  bargained,  sold  and  d^^'5 
these  presents,  do  bargain,  sell  and  deJa  5 
D.  C.  [Here  specify  tlie  pruvertu  sol  {'^j  " 
HOLD  the  aforesaid  bargained  prer 
C.  his  executoi's,  admiriisi:rafi)(70 


-.-'.^  the  said  II. /Hi 


12 
13 
14 
15 
16 
17 
}8 
19 
20 


200 
216  6 
235  3 
250 
9,6Q   6 
283  S 

00 
316 


162 
175 

187  5 


200 

212 
j225 
237 
250 


160 
173 

186  6 
200 
213  3 
226  6 
240 
253  3 
266  6 


7  1 

Sf278  5 

SCO 

321  4 

342  8 

;64 

>85  6 
407  1 

42B  5 


220 
240 

260 
280 
300 
320 
340 

;6o 

380 

0') 


200 

222  g 
244  4 


288  8 
311  1 

n  r^  r>     r 

355  5 
377  7 
400 

422  ^ 


143 

5 

164 

1 

184 

6 

205 

1 

225 

6 

246 

1 

266 

6 

287 

1 

307 

6 

328 

2 

348 

/ 

369 

0 

APPENDIX.  S39 

and  bequeatli  to  mj  dear  brother,  R.  A.  the  sum  of  ten 
pounds,  to  buy  him  mourning;.  I  give  and  bequeath  to 
my  son,  J.  A.  the  sum  of  two  hundred  pounds.  I  give 
and  bequeath  to  my  daughter,  E.  E.  the  sum  of  one  hun- 
dred pounds ;  and  to  my  daughter  A.  V.  the  like  sum  of 
one  hundred  pounds.  All  the  rest  and  residue  of  mj 
estate,  goods  and  chattels,  I  give  and  bequeatli  to  mj 
dear  beloved  wife,  E.  R.  whon*  I  nominate,  constitute 
and  apptnnt  sole  executrix  of  this  my  last  will  and  tes- 
tament, hereby  revoking  all  other  ancl  former  ^ilis  by  me 
at  any  time  heretofore  made.  In  witness  whereof,  I  have 
hereunto  set  my  hand  and  seal,  the  day  of 

ill  the  year  of  our  Lord 

Signed,  sealed,  publislied  and  declared  by  the  said 
testator,  B.  A.  as  and  for  his  last  will  and  testament,  in 
the  presence  of  us  who  have  subscribed  our  names  as  wit- 
/jesses  tliereto,  m  the  presence  of  the  said  testator. 

R.  A. 
S.  D. 
L.  T. 

Note.— The  testator  aRer  taking  off  his  seal,  must  ia 
presence  of  the  witnesses  pronounce  these  words, "  I  pub- 
lish and  declare  this  to  l>e  my  last  will  and  testament.'' 

Where  real  estate  is  devised,  three  witnesses  are  abso- 
lutely necessary,  who  must  sign  it  in  the  presence  of  the 
testator. 

A  LEASE  OF  A  HOUSE. 

KNOW  all  men  by  these  presents,  that  I,  A.  E.  of 
in  for  and  in  consideration  of  the  sum  of 

received  to  my  full  satisiaction  of  P.  V.  of 
this  day  of  in  tlie  year  of  our  Lord,        have 

demised  and  to  farm  let,  and  do  by  these  presents,  de- 
mise and  to  farm  let,  unto  the  said  P.  Y.  his  l.eirs,  execu- 
tors, admiTiistrators  and  asssigns,  on.!  certain  piece  of  land, 
lying  and  being  situated  in  said  bounded,  &g 

fHere  describe  the  boundaricsl  wiiii  a  dwelling-house 
tliereon  standing,  tor  the  term  of  one  year  from  this  date. 
To  HAVE  and  to  hold  to  him  the  -aid  i'.  V.  his  heirs, 
meoutors,  atimiui^trators  and  assigns  foi  sail  tci  nr,  lor 


f40  APPENDIX. 

bim  the  said  P.  V.  to  use  and  occupy,  as  to  him  shall  seem  meet 
and  proper.  And  the  said  A.  B.  doth  further  «ovenant  with 
the  said  P.  that  he  hath  good  right  to  let  and  demise,  the  said 
letten  and  demised  premises  in  manner  aforesaid,  and  that  he 
the  said  A.  during-  tlie  said  time  will  suffer  the  said  P.  quieUy 
lo  HAVE  and  to  HOLr,  use,  occupy  and  enjoy  said  demised  pre- 
•nises,  and  that  said  P.  shall  have,  hold,  use,  occupy,  possess 
and  enjoy  the  same,  free  and  clear  of  all  incumbrances,  claims, 
rights  and  titles  whatsoever.  In  witness  whereof,  I  tlie  said 
A.  B.  have  hereunto  set  my  hand  and  seal  this 
day  of 
ISignedj  sealed  and  delivered  )  AT? 

In  presence  (if  )  .  '     ' 

A  NOTE  PAYABLE  AT  A  BANK. 
[g500,  GO]  Hartford,  May  30,  1815. 

FOR  value  received,  I  promise  to  pay  to  John  Merchant, 
or  order,  Five  Hundred  Dollars  and  Sixty  Cents,  at  Hartford 
Bank,  in  sixty  davs  from  the  date. 

WILLIAM  DISCOUNT. 


AN  INLAND  BILL  OF  EXCHANGE. 

[g83,  34]  Boston,  June  1,  1815. 

TWENTY  days  after  date,  please  to  pay  to  Thomas 
Goodwin  or  order,  Eighty-Three  Dollars  anil  Thirt}'"-Four 
Cents,  and  place  it  to  my  account,  as  per  advice  from  your 
humble  servant,  SIMON  PURSE. 

Mr.  T.  W.  Merchani,  } 
JVexV'York.  J 

A  COMMON  NOTE  OF  HAND. 

[;J1303  New- York,  March  8, 1821. 

FOR  value  received,  I  promise  to  pay  to  John  Murray, 
One  Hundred  and  Thirty  Dollars,  in  four  months  from  this  date, 
with  interest  until  paid.  JOHN  LAWRENCE. 

A  COMMON  ORDER. 

NEw-Yor     June  10»  i822 

JSlr.  CJuirlcs  Careful^ 

P>Rase  to  deliver  Mr.  George  Speedwell,  the  amount  di 
Twenty -Five  Dollars,  in  goods,  from  your  store;  and  charge 
the  same  to  the  account  of  Your  ObH.  Servant, 

E.  WHITE. 
FiNIB. 


THE 

PRACTICAL  ACCOUiTTANT, 

OR, 

BEST  METHOD  OP 

FOa  THE 

EASY  INSTRUCTION  OP  YOUTH. 

DESIGNSD 

AS  A  COMPANION  TO  DABOLL^S 

ARITHMETIC. 


BY  SAMUEL  GBEEN. 
KfBUSHED  BY  SAMUEL  GKEEN. 

ICBVV',L0N2iDK 


INTRODUCTION. 


Scholars,  male  and  female,  after  they  have  acquired  a 

*  rit  kuowledg-e  of  Arithmetic,  especially  in  the  funda* 

!  rules  of  Addition,  Subtraction,  Multiplication,  and  Di- 

1.,  should  be  instructed  in  the  practice  of  Book  Keeping, 

v  this  it  is  not  meant  to  recommend  that  the  son  or  daughter  of 
V  very  farmer,  mechanic,  or  shop  keeper,  should  enter  deeply 
into  the  science  as  practised  by  the  merchant,  engaged  in  exten- 
sive business,  for  such  study  would  eng-ross  a  great  portion  of 
time  which  might  be  more  usefully  employed  in  acquiring  a 
proper  knowledge  of  a  trade,  or  other  employment. 

Persons  employed  in  the  comiton  business  of  life,  who  do  not 
^leep  regpilar  accounts,  are  subjected  to  many  losses  and  incon- 
veniences; to  avoid  which,  the  following  simple  and  conreot 
plan,  is  recommended  for  tlieir  adoption. 

Let  a  small  book  be  made,  or  a  iew  sheets  of  paper  sewed 
together,  and  ruled  after  the  examples  given  in  this  system.  In 
the  book,  termed  the  Day  Book,  are  duly  to  be  entered,  daily, 
all  the  transactions  of  ths  master  or  mistress  of  the  family,  which 
require  a  charge  to  be  made,  or  a  credit  to  be  given  to  any  per- 
ion.  No  article  thus  subject  to  be  entered,  should  on  any  con- 
sideration, be  deferred  till  another  day.  Great  attention  should 
be  given  to  write  the  transaction  in  a  plain  hand ;  the  entry 
■should  mention  all  the  particulars  necessary  to  make  it  fully  un- 
derstood, with  the  time  when  they  took  place  ;  and  if  an  article 
^e  delivered,  the  name  of  tlie  person  to  whom  delivered  is  to  be 
mentioned.  No  scratching  out  may  be  suffered ;  because  it  in 
sometimes  done  for  dishonest  purposr  j,  and  will  weaken  or  de- 
stroy the  authority  of  your  accounts.  But  if,  through  mistake, 
any  transaction  should  be  wrongly  entered,  the  error  must  be 
rectified,  by  a  new  entry :  and  the  wrong  one  may  be  cancelled 
by  writing  the  word  Error ^  in  the  margin. 

A  book,  thus  fairly  kept,  will  at  all  times  show  the  exact  state 
of  a  person's  affairs,  and  have  great  weight,  should  there  at  any 
tine  be  a  necessity  of  producing  it  in  a  court  of  Justice. 


FORM  OF  A  DAY  BOOk'. 


*  JEREMIAH  GOODALE,  Alhviu,  January  1, 


Entered. 

1 


Joseph  Mii5tii>?vs, 
By  3  montlis'  wages,  at  ^o  i 
(late,         .         . 


Entered.      Samuel  Stacy, 

1  'To  2  weeks'  wa^R  «"' 
spinning  \^ri\,  at  7^ 
tiiisday, 


Entered. 

1 


Entered. 
1 

Entered. 

1 


Entered. 

1 


Entered. 

1 


Anthony  Biliinj^s,  .         .  Cr. 

By  my  order  in  favor  of  Joseph  Hasting-s, 

15 


Joseph  liastiii^;^,    .         .         .         Dr. 
To  my  order  for  goods  out  of  the  store  of 
Anthony  Billings, 


Thomas  Grosvenor,         .         .        Dr. 
To  the  frame  of  a  house  completed  and  raised 
this  day  on  his  Glover  Fann,  so  called, 
4000  feet  at  2.^  cents  per  foot, 

1 8 _.! 


Edward  Jones,         .         .         .        Cr. 
By  his  team  at  sundiy  times,  carrying  ma- 
nure on  my  farm,         .         .         .         .    ii 

25 • 


Entered. 
1 


filtered. 
1 


Thomas  GrosvenOr,         .         .         Dr. 
To  48  window  sashes  delivered  at  his  Glover 
Farm,  so  called,  at  g  1,00,   .       .     ^48,00 
Setting  500  panes  of  glass  by  my  son 

John,  at  1  i  cents,  .         .  7,50 

10  days'  work  of  myself  finishing  front 

room,  at  ^1,25  a  day,     .         .       12,50 
7i  do.  of  William,  my  hired  man,  ^ 

laying  tlif  kitchen  floor  and  hang-  >  6,30 

ing  doors,  at  84  cents  a  Jay,           )  — 
26-— 


Anthony  Billings,       .         .         .     Cr. 

By  2  galls,  molasses  at  33  cts.  pergail.  0-,72 

*4  yds.  of  India  Coljon,  at  18|fcents,  .0,74 

2  flannel  sMrts  to  Joseph  Hastings      2J6 


Joseph  Hastings, 
To  2  shirts  of  A.  Billings, 


Dr. 


II  T 


«  Thtrt  pvt  the  name  of  the  owur  <f  the  iofJc.  andjirsi  date. 


"Entered. 
1 

Entered. 

1 

Entered. 
1 


Entered. 
1 


jBntered. 
1 

Entered. 
1 

Entered. 
1 


Entered. 
1 


FORM  OF  A  DAY  BOOK. 

Albany,  February  V2,  18^2. 


Joseph  Hasling"s, 
To  my  order,  on  T.  Grosvenor, 

16 


Thomas  Grosvenor,         .         .       *T>v. 
To  3  days'  work  of  myself  on  yorrr  fence 
atgl,25perdaT,         .      " .         .      3,75 
3  days'  do.  my  man  Wm.  on  your  stable 
and  finisliing  off  kitchen,  at  84  cts.     2,52 
2  pr.  brown  yam  stocldngs,  at  4*2  cts,  0,84 

^ .    _1 


Edward  Jones,         .         .         .       Cr 
By  4  months'  hire  of  his  son  William  at  glO 
a  month, 


Edward  Jones,         .         .         .       Dr 
To  my  draft  on  Thomas  Grosvenor, 


Lntered. 
1 


Thoinas  Grosvejior,         .         .         Cr. 
By  my  order  in  frv'or  of  Joseph  Hastings, 


©r. 


"^4« 


Thomas  Grosrenor, 
By  my  draft  in  fafor  of  E.  Jones, 

-.28 — ■' 


Cr. 


Tliomas  Grosvenor 
To   t-ie  frame  cf  h  barn 


Dr. 


Anthony  Billings,  .  .         Cr. 

For  the  following  articles, 
14  lbs.  muscovado  sugar  at  ^12  prc"*vt  1,50 


1  larg-e  dish, 

6  plates, 

4  cups  and  saucers, 

1  pint  French  Bmudy, 

1  quart  Cherry  Bounce, 
Thread  and  ta]:  e,    . 

2  Thimbles, 
1  pair  Scissors,        . 
1  quire  p-Aper, 
Wafers,  4  ;  ink,  6  ;  1  bottle,  8  ; 


0,23 
0,30 
0,20} 
0,17, 
0,33 
0,1$ 
0,04 
0,17 
0,25 
0,18 


Peter  Dciboll,         .         .         .        Dr.  ' 

To  a  cotton  Coverlet  delivered  Sai-ah  Brad- 

fortij  bv  vonr  wf Kten  order,  dated  14,  Jan. 


tO'RM  OF  A  DAY  BOGK. 


1 

Entered. 
1 


Entered. 

1 


Entered. 
1 

Entered. 
1 


Entered. 
2 


Albany,  Marchl^l822. 


Thomas  Grosvenor,        .         .        Cr. 
By  1  b^el  containing  cider  sold  and  deliv- 
ered to  Anthony  Billings,     . 
10 


Thomas  Grosvenor, 
B"  cash  paid  me  this  date, 

' 4 


Cr. 


Anthony  Billings,  .         .  Dr. 

To  one  Barrel  of  Cider,       .         .        jl,17 

1  baiTel  containing  the  same  (from 

Thomas  Grosvenor,)      .         .         .  0,58 


Anthony  BiUings,       .         .         .     Dr. 
To  casii  per  his  order  to  George  Gilbert, 
: — 1 5 ., 


Cr. 


Peter  Dabo?!, 
By  amount  of  his  shoe  account,     .      §4,48 
Yam  received  from  him  for  the  bal- 
ance of  his  account,       .         .         .1,03 


Samuel  Green, 


Cr, 


By  amount  dne  for  12  months  New- 
London  Gazette,       .         .         .      ^2,00 
4  SpeUmg  books  at  20cts.  for  chil- 
dren,        .         .         .         .         .    0,80 

1  Daboil's  Arithmetic,  for  my  son 
Samuel,      ....  0,42 

2  Blank  Writing  books  at  1 2^  cents,   0,25 
1  quire  of  Letter  Paper,       .         .    0,34 


Eutcred. 

2 


Entered. 
2 


Cntered. 


-24- 


Notes  pavable,  .         .       Cr. 

By  my  note  ol  this  data  trri,loi*sed  by  Ephraim 
Dodge,  ut  6  .i'o;%tlis,  for  a  yoke  of  Oxen 
bought  of  Daniel  Mason,  at  Lebanon, 

, 28 


Jonathan  Curtis,       .         .         .      Dr,      i 
To  an  old  bay  horse,         .         .         ^23,Oo!i 
a  four  vrheeled  waggon,  and  lialf 

worn  harness,        .         .         •       42,00 


I    Samuel  Green, 
To  cash  in  full, 


Dr. 


rORM  OF  A  DAY  BOQK. 


ESered. 


Albany,  April  8,  1822. 


Entered. 
1 

Entered 
1 


Entered. 

2 

Entered. 
1 


Anthony  Bill  ing-s,         .         .         .  Dr. 
To  2  tons  of  Hay  at  gl  1,25,     .       .  g22,50 

Amount  of  order  dated  March  26  th,  ) 
1822,  in  favour  of  Fanny  White,  >  0,54 
paid  in  1  pair  yam  stocking,        ) 

Hire  of  my  wag-g-on  and  horse  to  i 
bring  sundry  articles  from  Provi-  ^  3,00 
dence,  3d  of  this  montii,       .        ) 

12 ■ 


Entered. 

2 

Entered. 
1 


fotered. 


Thomas  Grosvenor,      .         .       .  Cr. 
By  his  order  on  Theodore  Barreil,  New- 
London  for  68  dollars,       . 


Anthony  Billings, .     .         .         .     Dr. 

To  1  hogshead  Rum  from  Theodore  Barrell, 

100  galls,  at  50  cents,       .       .  g50,00 

Cash  received  from  said  Ban-ell  for 
balance  due  on  Thomas  Grosve- 
nor's  order,       .       ♦         .       .       1 8,00 


— 18- 


Jonathan  Curtis,       .         .         .      Cr. 
By  a  coat  ^M,75,  pantaloons  ^5,00, 

■ .^ 22 


Thomas  Grosvenor,         .         .         Dr. 
To  mending  your  cart  by  my  man  Wil- 
liam,        gl,00 

Paid  Hunt,  for  blacksmith's  work  on 
your  cart,        .         .         .         .0,5.8 

Setting  6  panes  of  glass,  and  finding 
fflass,       .         .         .         .         .      0,66 


-25- 


John  Rogers,     .         .         .         .     Dr. 
To  a  voke  of  oxen,  at  60  days'  Credit, 


Anthony  Billings,       .         .         .     Cr. 
By  gardeti  seeds  of  various  kinds,       $0,56 

1  pair  of  boots,  myself,  §4,00,  and  1 
pah-  for  John,  §J3,50,  .         .     7,50 

1  pair  of  thick  shoes  for  Joseph  Has- 
tings,      .         .         .         .         .      1,25 

Tea,  Sugar,  and  Lamp  Oil,  per  bill,  0,68 


Notes  payable, 
By  my  note  to  Isaac  Thompr 


rORM  OF  -A  r>AY  KOOIC. 


Albciiiy,  May  3,  1(^22, 


Thecxiorc  JbcUToU,  Nev/-London,     Dr. 
To  16  cheeiie,  308  ibs.  at  5  cents,       g  1 5,40 
217  lbs.  of  butter,  at  15  2-'3  cents,    34,00 
24  lbs.  of  honey,  at  12^  cents,  3,00; 


Entered. 


Entered. 
1 


Entered. 
1 


Entered. 
1 

Entered. 


Batercd. 
1 

Entered. 


Entered. 

2 


Joseph  Hastings,    .         .         •         Dr. 
To  1  pair  shoes,  2yth  April,  f:..;m  Antimony 

Biliin?^, 

12 — 


Anthony  Billing's,     .         .         .Dr. 
To  84  bushels  of  seed  potatoes,  at  33  1-3 

cetiLs,         .         .  .       g28,G0 

8  pair  rnittcns  at  20  cents,       .       .1,60 

Cash, 14,00 

•15- 


Joseph  Hastings, 
By  4h  months  -vf  o^-es  at  7  dollars, 

iO 


Cr. 


Theodore  Bari-oil, 
By  cash  in  full  of  all  demands, 


Or. 


Thomas  Grosvenoi*,  .  .  Cr. 
By  his  acceptance  of  ray  order  in  favor  of 

Anthony  Billinj^s, 

Anthony  Biihngs,     .         .  Dr. 

iToamount  of  mv  order  on  Thomas  Grosrc- 


-.Sept.  24- 


'  Notes  payable,         .         .         .       Dr. 
To  cash  paid  for  my  nolo,  to  D.  r.Iason, 


C. 


52 


3 
62 


40 


25 


60 
0 

40 


54  00 


54  00 


48  loo 


'ln&  foregoiijg  exaajplc  of  a  Day  Bckjk.  r»iay  suince  to  give  a  g<jod  idea  of 
the  way  in  which  it  i3  p'-oper  to  make  the  original  entries  of  all  debt  and 
credit  articles.  Anotber  small  book  should  next  be  prepared,  according  to 
the  foiiowmg  form,  termed  the  book  of  Accounts,  or  Leger.  into  this  book 
must  be  posted  the  whole  contents  of  the  iM^y  Book ;  care  being  taken  that 
every  article  be  carried  to  its  correspondiuix  title;  tlie  debt  amounts  to  be 
entered  iii  the  left,  and  the  credit  in  the  right  hand  page.  Thus,  should  it 
lit  any  time  be  required  to  know  the  ^late  of  an  accouiit,  it  will  only  be  ne- 
cessary to  sum  up  the  two  columns,  and  to  subtract  the  smaller  amount 
from  the  greater,  the  remainder  wili  be  the  balance. 

When  an  ar<icle  is  posted  from  the  Day  Book  into  the  Leger.  it  will  be 
proper,  opposite  the  article,  to  nole  the  «ame  in  the  margin  of  the  Day  Book, 
by  writing  the  worrt  Entered^  or  making  two  parallel  stiokes  with  the  pen- 
to  which  should  be  added  the  figure  deaoting  the  page  in  the  Leger,  whera 
the  account  is. 

On  a  blank  page  at  the  beginning,  or  end  of  the  Leger,  an  aiphahetcal 
index  shoulcl  be  written,  containing  th?  muxfe^  ef  every  pereoa  v^'ik  ^^r^KKM 
%aa  have  accounts,  in  the  Leger,  with  t»  B^aah^  eC  IM  ^«f*  W9iik  ^ 
;)Ccounts  are. 


Dr. 


FORIVf  OF  A  LEGEKr 

Joseph  Hastings, 


1822. 

Fcb'y. 
May 


To  my  order  on  Anthony  Billings  for  gooids, 
2  shirts  of  Anthony  Billings, 
My  order  on  Thomas  Grosvenor, 
1  pair  shoes,  29th  April,  from  A.  Billing^s, 


2|18 


50 


Djt.                     Samuel  Stacy, 

1022.  J                                                                                 1  g  O 

Jap.''y.j  6  To  2  weeks'  wages  of  my  daughtep  at  75   )     -  ^^ 

1             cents  a  week,       .           .          .        ,      \ 

Dr.                    Anthonj  Billings, 

1822. 
March 

April 

May 


4  To  1  barrel  of  cider  and  barrel, 
10     Cash  paid  your  order  in  favor  of  G.  Gilbert 

6      Sundries,      ,...'.. 
12         ditto.    .         .         -         .    '     . 

12         ditto 

25]    My  order  on  Thomas  Grosvenor, 


75 


32 
04 


68  00 


60 
00 


Dr. 


Thomas  Grosvenor, 


1822. 
Jan'y. 

Feb?y. 

April 


To  the  frame  of  a  house, 
Sundries, 
Sundries, 

The  frame  of  a  bam. 
Sundries, 


IlOOf 
i  74^0 

7,U 
75.00 

I  2m 


Dr. 


1822.  I 
Feb'y.  24 


Edward  Jones, 


To  my  draft  on  Thomas  Grosvenor, 


i  38  oe 


Dr. 


Eater  Daboil, 


1822.   I 

F^b'j.  1 28(Tci  suirtlrffes* 


if' 


FQMI  O?  A  LEGEK, 


A  \nre6  !af! 


Farmer, 


Cr. 


1822, 
JaaV. 
May 

1 
15 

By  3  iiiuiilhs'  v/a;;'^'^  dii'j  ihi^  tiL.y  2.1  ^^, 
4i  mciitlis'  \Ysges  at  g7, 

18 
31 

C. 

00 
50 

Cr. 


Merchuiit, 


Cr 


1822.  I     I 

Jan'}%  I  5  IB  J  my  order  in  faror  of  Joseph  Hastmgs, 

2G      Sundries, 
Feb'y.NS         ditto. 
April    29  ditto 


C. 

50 

62 
55 
00 


Judge  of  County  Court, 


Cn 


Fv.  '22 

12 

By  my  order  in  favor  of  Joseph  Hastings, 

g3|50 

ss'ao 

24 

My  draft  in  favor  of  Edward  Jones, 

March 

1 

Cash  paid  me  this  day, 

75  OO 

1  en^pty  cider  bairel,     . 

5» 

April 

12 

Amount  of  \  ourordcr  on  Theodore  Barrell, 

68  00 

May 

25 

My  order  in  favor  of  Anthony  Billing-s, 

54  00 

Labourer, 


Cr. 


1822.   I 

Jan'y*  |l<>ii^y  team  hire  at  sundry  limes, 

Feb'y. II 8 1     4  mon ths'  lure  of  his  son  WiUiam  at  g  1 0, 


5  64 
4o!ort 


Farmer, 


Cr. 


1822.  I     I 

MarchUaJBy  sundries  in  full> 


■.      I     5161 


w 


« 

FOMT  OF  A  LEGER. 

Dr.                       Samuel  Green, 

Mar. 

28  To  cash  in  full  of  hh  occount, 

3  81 

Dr.                      Notes  Payable, 

Sept.   !'24'To  cash  psJd  for  mf  note  to  D.  Mascm, 


May    I  31  To  16  cheese,  weight  308  lbs.  at  5  cents, 
\     I     217  lbs.  butter  at  15  2-3  cents,      .       . 
\     j    24  lbs.  honey  at  12^  cents, 


% 

4f! 


Dr.                     Jonatnan  Curtis, 

1822.   j 

Blarcb!28  To  abay  liorse, 

i         A  wagg-on  aud  harness,       .       .       •      . 

23  00 

42  00 

Dr.                       John  Rogers, 

1822.  1 

ilpril    25  To  1  yoke  of  oxen  at  60  days'  credit, 
1 

i  c. 

60  0.0 

Dr.                   Theodore  Barrel!, 

34 


52 


INDEX  TO  THE  LEGER. 


RarrcU  Theodore, 

PAGE. 

.    2 
.    1 

H. 

Hastings  Joseph,   . 

PAG^i 

.       .    1 

Billings  Anthony, 

J. 

Jones  Edward,    , 

C. 

.    2 

1 

Curtis  Jonathan,  . 

N. 
Notes  Payable,    . 

D. 

.    1 

•        .    ^ 

DaboU  Peter, 

R. 

Rogers  John, 

G. 

.    1 
.    2 

.    0 

Grosvenor  Thomas, 
Chrten  Samuel, 

B. 
Stacy  Samnol,     . 

.       .    t 

FORM  Ot^  A  l.EGfiR 


New  London, 

Cr. 

1822. 
March 

U 

By  sunJnes,      .                 ... 

3  81 

Cr. 

/822. 
March 

April 

24[By  »y  note  to  Daniel  IVIason,  at  6  months, 

endorsed  by  Ephraim  Dodge, 
29      Do.  Isaac  Thompson,  at  6  months, 

48 
90 

c. 

00 
00 

Danbury, 

Cr. 

1822. 
April 

1     1 

IBjByacoat, 

1    A  pair  of  pantaloons. 

if 

1473 

5[oa 

Hudson, 

Cr. 

1  i 

1  I 

i\o 

New  London, 


Cr. 


1822. 
IM^y 


20 


By  cash  m  full, 


52 
521 


Cv 
40 

40 


QUESTIONS  TO  EXERCISE  THE  STUDENT. 


What  is  the  stale  of  the  following  Accounts, 


Joseph  Hastings, 
Samuel  Stacy, 
Anthony  Billing, 
Thomas  Grosvenor, 
Eflward  Jones, 
Notes  Payable, 
Jbnathrtu  Curtis^ 


Due  Joseph  Hastings, 

^Edward  Jones,  . 

^Notes  Payable,  . 

Samuel  Stacy  owes, 
I  Anthony  Billings  owes, 
Thomas  Grosvenor  owes, 
(Jonathan  Curtis  owes, 


g31,0» 

7,64 

90,00 

1,50 

189,05 
19, 6*^ 

45,25 


•&  Farmer^s  ThUj  or  Acxmml. 

Auburn,  Oct.  21,  t8a«. 
^omas  Yates,  Esq. 

To  John  Mominaton,  Dr. 

April  5.     To  5  barrels  Cider,  af          g2,00      .       .  g  10,00 

20  bushels  Potatoes,  at    0,25  .       •       .  5,00 

55  lbs.  Butter,  at              0,17      .       .  9,35 

June  6,       ■     1  ton  of  Hay,        .        .        ...  10,00 

Juljr  15.         40  lbs.  Cheese,  at             0,08      .       »  3,20 

2  cords  of  Wood,  at       4,00  .      .       .  8,00 

Received  the  amount.  lS^"?j^5 

JOHN  MORNINGTON. 
N.  B.-5rTo  prevent  accidents,  care  should  bs  taken  not  to  re* 
ceipt  an  account  until  it  is  paid. 

A  negotiahle  J^oie, 

New-Haven,  March  21,  1822. 
Six  monthjs  after  date,  I  promise  to  pay  to  William  Walter,  o? 
order,  (at  my  house,)  One  Hundred  Dollars,  value  received  in 
imo  yoke  of  oxen.  JAMES  HILI^IOUSE, 

Qi^^It  is  best  to  mention  where  the  note  sliiall  be  paid,  and  for 
what  it  is  given.     Without'  the  words,  "  or  order^''  a  note  v. 

not  negotiable.  ■ 

A  Receipt  in  full, 
Received,  Hartford,  Mav  2^2, 1822,  of  Theodore  Barrell,  Esq. 
Fifty-two  DoUars  in  full  of  all  demands.    GEO.  GOODWIN. 
(t^If  the  payment  be  not  hi  full,  write  "  on  accounV^ 
N.  B. — For  other  useful  foims  see  the  Arithnretic. 


.VOTE. 

The  affectionate  Instructer,  who  alp/ajs  feels  a  parental  soli- 
citude for  the  permanent  welfare  of  his  pupils,  cannot  m  any  way 
8o  much  contribute  to  their  success  in  life,  witii  so  little  trouble, 
as  to  teach  them  to  understand  this  abridged,  complete  and  sim- 
ple system  of  Book  Keeping*.  It  contains  all  the  important 
principles  of  extended  and  expensive  works  on  the  science ;  alt,^ 
in  fact, that  is  necessary  to  he  loiov/n  by  the  Farmer,  Mechan- 
ic, and  Shopkeeper,  relating"  to  accounts ;  and  yet  with  very 
little  explanation  and  repeated  copying*  and  balancing  tlie  ac- 
counts, will  fee  so  fully  understood  and  deeply  impressed  on  ihd 
memory  of  sckolars  of  common  mind,  as  never  to  be  forgotten; 
wiiile  tteir  knowledge  of  common  arithmetic  and  practical 
S^empHMhip  wiil  tliwcby  be  greatly  improved. 
FINIS' 


1 


VA- 03572 


^^ 


■fe 


I 


Ivilll475 


i^. 


1^% 


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